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Boundless Lecture Slides

Boundless Lecture Slides. Available on the Boundless Teaching Platform. Free to share, print, make copies and changes. Get yours at www.boundless.com. Using Boundless Presentations. Boundless Teaching Platform

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Boundless Lecture Slides

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  1. Boundless Lecture Slides Available on the Boundless Teaching Platform Free to share, print, make copies and changes. Get yours at www.boundless.com

  2. Using Boundless Presentations Boundless Teaching Platform Boundless empowers educators to engage their students with affordable, customizable textbooks and intuitive teaching tools. The free Boundless Teaching Platform gives educators the ability to customize textbooks in more than 20 subjects that align to hundreds of popular titles. Get started by using high quality Boundless books, or make switching to our platform easier by building from Boundless content pre-organized to match the assigned textbook. This platform gives educators the tools they need to assign readings and assessments, monitor student activity, and lead their classes with pre-made teaching resources. Get started now at: • The Appendix The appendix is for you to use to add depth and breadth to your lectures. You can simply drag and drop slides from the appendix into the main presentation to make for a richer lecture experience. http://boundless.com/teaching-platform • Free to edit, share, and copy Feel free to edit, share, and make as many copies of the Boundless presentations as you like. We encourage you to take these presentations and make them your own. If you have any questions or problems please email: educators@boundless.com Free to share, print, make copies and changes. Get yours at www.boundless.com

  3. About Boundless • Boundless is an innovative technology company making education more affordable and accessible for students everywhere. The company creates the world’s best open educational content in 20+ subjects that align to more than 1,000 popular college textbooks. Boundless integrates learning technology into all its premium books to help students study more efficiently at a fraction of the cost of traditional textbooks. The company also empowers educators to engage their students more effectively through customizable books and intuitive teaching tools as part of the Boundless Teaching Platform. More than 2 million learners access Boundless free and premium content each month across the company’s wide distribution platforms, including its website, iOS apps, Kindle books, and iBooks. To get started learning or teaching with Boundless, visit boundless.com. Free to share, print, make copies and changes. Get yours at www.boundless.com

  4. Differential Equations Differential Equations, Parametric Equations, and Sequences and Series Parametric Equations and Polar Coordinates Infinite Sequences and Series ] Differential Equations, Parametric Equations, a... Free to share, print, make copies and changes. Get yours at www.boundless.com

  5. Differential Equations, Parametric Equations, a... > Differential Equations Differential Equations • Solving Differential Equations • Models Using Differential Equations • Direction Fields and Euler's Method • Separable Equations • Logistic Equations and Population Grown • Linear Equations • Predator-Prey Systems Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus/textbooks/boundless-calculus-textbook/differential-equations-parametric-equations-and-sequences-and-series-4/differential-equations-16/

  6. Differential Equations, Parametric Equations, a... > Parametric Equations and Polar Coordinates Parametric Equations and Polar Coordinates • Parametric Equations • Calculus with Parametric Curves • Polar Coordinates • Area and Arc Length in Polar Coordinates • Conic Sections • Arc Length and Speed • Conic Sections in Polar Coordinates Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus/textbooks/boundless-calculus-textbook/differential-equations-parametric-equations-and-sequences-and-series-4/parametric-equations-and-polar-coordinates-17/

  7. Differential Equations, Parametric Equations, a... > Infinite Sequences and Series Infinite Sequences and Series • Sequences • Series • The Integral Test and Estimates of Sums • Comparison Tests • Alternating Series • Absolute Convergence and Ratio and Root Tests • Tips for Testing Series • Power Series • Expressing Functions as Power Functions • Taylor and Maclaurin Series • Applications of Taylor Series • Summing an Infinite Series • Convergence of Series with Positive Terms Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus/textbooks/boundless-calculus-textbook/differential-equations-parametric-equations-and-sequences-and-series-4/infinite-sequences-and-series-18/

  8. Appendix Free to share, print, make copies and changes. Get yours at www.boundless.com

  9. Differential Equations, Parametric Equations, a... Key terms • accelerationthe change of velocity with respect to time (can include deceleration or changing direction) • analytic functiona real valued function which is uniquely defined through its derivatives at one point • analytic functiona real valued function which is uniquely defined through its derivatives at one point • boundary conditionthe set of conditions specified for behavior of the solution to a set of differential equations at the boundary of its domain • Cauchy sequencea sequence whose elements become arbitrarily close to each other as the sequence progresses • combinatoricsa branch of mathematics that studies (usually finite) collections of objects that satisfy specified criteria and their structures • complex analysistheory of functions of a complex variable; a branch of mathematical analysis that investigates functions of complex numbers • conditional convergenceA series or integral is said to be conditionally convergent if it converges but does not converge absolutely. • conea surface of revolution formed by rotating a segment of a line around another line that intersects the first line • conea surface of revolution formed by rotating a segment of a line around another line that intersects the first line • convergeof a sequence, to have a limit • convergence testmethods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence, or divergence of an infinite series Free to share, print, make copies and changes. Get yours at www.boundless.com

  10. Differential Equations, Parametric Equations, a... • coordinatea number representing the position of a point along a line, arc, or similar one-dimensional figure • curvea simple figure containing no straight portions and no angles • decayTo change by undergoing fission, by emitting radiation, or by capturing or losing one or more electrons. • definite integralthe integral of a function between an upper and lower limit • derivativea measure of how a function changes as its input changes • derivativea measure of how a function changes as its input changes • derivativea measure of how a function changes as its input changes • differentiablehaving a derivative, said of a function whose domain and co-domain are manifolds • differentiablehaving a derivative, said of a function whose domain and co-domain are manifolds • differential equationan equation involving the derivatives of a function • differential equationan equation involving the derivatives of a function • differential equationan equation involving the derivatives of a function Free to share, print, make copies and changes. Get yours at www.boundless.com

  11. Differential Equations, Parametric Equations, a... • differential equationan equation involving the derivatives of a function • differential equationan equation involving the derivatives of a function • displacementa vector quantity which denotes distance with a directional component • fractiona ratio of two numbers, the numerator and the denominator; usually written one above the other and separated by a horizontal bar • functiona relation in which each element of the domain is associated with exactly one element of the co-domain • hyperbolaa conic section formed by the intersection of a cone with a plane that intersects the base of the cone and is not tangent to the cone • improper integralan integral where at least one of the endpoints is taken as a limit, either to a specific number or to infinity • improper integralan integral where at least one of the endpoints is taken as a limit, either to a specific number or to infinity • improper integralan integral where at least one of the endpoints is taken as a limit, either to a specific number or to infinity • integral testa method used to test infinite series of non-negative terms for convergence by comparing it to improper integrals • limita value to which a sequence or function converges • limit superiorthe supremum of the set of accumulation points of a given sequence or set Free to share, print, make copies and changes. Get yours at www.boundless.com

  12. Differential Equations, Parametric Equations, a... • linear equationa polynomial equation of the first degree (such as ) • monotoneproperty of a function to be either always decreasing or always increasing • natural logarithmthe logarithm in base • non-linear differential equationnonlinear partial differential equation is partial differential equation with nonlinear terms • normalize(in mathematics) to divide a vector by its magnitude to produce a unit vector • polarof a coordinate system, specifying the location of a point in a plane by using a radius and an angle • polarof a coordinate system, specifying the location of a point in a plane by using a radius and an angle • power lawany of many mathematical relationships in which something is related to something else by an equation of the form • predatorany animal or other organism that hunts and kills other organisms (their prey), primarily for food • preya living thing that is eaten by another living thing • Pythagorean TheoremA theorem stating that the hypotenuse of a right triangle is equal to the square root of the sum of the square of the other two sides • recursionthe act of defining an object (usually a function) in terms of that object itself Free to share, print, make copies and changes. Get yours at www.boundless.com

  13. Differential Equations, Parametric Equations, a... • sequencean ordered list of objects • sequencean ordered list of objects • seta collection of distinct objects, considered as an object in its own right • simultaneous equationsfinite sets of equations whose common solutions are looked for • summandsomething which is added or summed • tangenta straight line touching a curve at a single point without crossing it there • trajectorythe path of a body as it travels through space • Z-transformtransform that converts a discrete time-domain signal into a complex frequency-domain representation • Zeno's dichotomyThat which is in locomotion must arrive at the half-way stage before it arrives at the goal. Free to share, print, make copies and changes. Get yours at www.boundless.com

  14. Differential Equations, Parametric Equations, a... Polar Coordinates A set of polar coordinates. Note the polar angle increases as you go counterclockwise around the circle with 0 degrees pointing horizontally to the right. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Polar coordinate system."CC BYhttp://en.wikipedia.org/wiki/Polar_coordinate_system%23ConventionsView on Boundless.com

  15. Differential Equations, Parametric Equations, a... Linear equations Graphical example of linear equations. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Linear equations."CC BYhttp://en.wikipedia.org/wiki/Linear_equationsView on Boundless.com

  16. Differential Equations, Parametric Equations, a... Exponential Function as a Power Series The exponential function (in blue) and the sum of the first terms of its Taylor series at (in red) up to . Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Taylor series."CC BYhttp://en.wikipedia.org/wiki/Taylor_seriesView on Boundless.com

  17. Differential Equations, Parametric Equations, a... Zeno's Paradox Say you are working from a location toward . Before you can get there, you must get halfway there. Before you can get halfway there, you must get a quarter of the way there. Before traveling a quarter, you must travel one-eighth; before an eighth, one-sixteenth; and so on. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Zeno's paradoxes."CC BYhttp://en.wikipedia.org/wiki/Zeno's_paradoxesView on Boundless.com

  18. Differential Equations, Parametric Equations, a... Integral Test The integral test applied to the harmonic series. Since the area under the curve  for is infinite, the total area of the rectangles must be infinite as well. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Integral test for convergence."CC BYhttp://en.wikipedia.org/wiki/Integral_test_for_convergenceView on Boundless.com

  19. Differential Equations, Parametric Equations, a... in Taylor Approximations Figure shows and Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13. As more power functions with larger exponents are added, the Taylor polynomial approaches the correct function. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Taylor series."CC BYhttp://en.wikipedia.org/wiki/Taylor_seriesView on Boundless.com

  20. Differential Equations, Parametric Equations, a... Trajectories A trajectory is a useful place to use parametric equations because it relates the horizontal and vertical distance to the time. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Trajectory."CC BYhttp://en.wikipedia.org/wiki/TrajectoryView on Boundless.com

  21. Differential Equations, Parametric Equations, a... Alternating Harmonic Series The first fourteen partial sums of the alternating harmonic series (black line segments) shown converging to the natural logarithm of 2 (red line). Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Harmonic series (mathematics)."CC BYhttp://en.wikipedia.org/wiki/Harmonic_series_(mathematics)View on Boundless.com

  22. Differential Equations, Parametric Equations, a... Infinite Series An infinite sequence of real numbers shown in blue dots. This sequence is neither increasing, nor decreasing, nor convergent, nor Cauchy. It is, however, bounded. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Sequence."CC BYhttp://en.wikipedia.org/wiki/SequenceView on Boundless.com

  23. Differential Equations, Parametric Equations, a... Arc Length The curved lines bounding the region are arcs. Their length can be calculated with calculus. The area of the region can also be calculated by integration. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Polar coordinate system."CC BYhttp://en.wikipedia.org/wiki/Polar_coordinate_system%23ConventionsView on Boundless.com

  24. Differential Equations, Parametric Equations, a... Types of Conic Sections There are three types of conic sections: 1.Parabola; 2. Ellipse; 3. Hyperbola. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Conic section."CC BYhttp://en.wikipedia.org/wiki/Conic_sectionView on Boundless.com

  25. Differential Equations, Parametric Equations, a... Example slope field The slope field of , with the blue, red, and turquoise lines being , , and , respectively. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Slope field."CC BYhttp://en.wikipedia.org/wiki/Slope_fieldView on Boundless.com

  26. Differential Equations, Parametric Equations, a... Euler's Method Illustration of the Euler method. The unknown curve is in blue and its polygonal approximation is in red. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Euler method."CC BYhttp://en.wikipedia.org/wiki/Euler_methodView on Boundless.com

  27. Differential Equations, Parametric Equations, a... Logistic Curve The standard logistic curve. It can be used to model population growth because of the limiting effect scarcity has on the growth rate. This is represented by the ceiling past which the function ceases to grow. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Logistic function."CC BYhttp://en.wikipedia.org/wiki/Logistic_functionView on Boundless.com

  28. Differential Equations, Parametric Equations, a... Relation between Cartesian and Polar Coordinates The Cartesian coordinate is given by and the Cartesian coordinate is given by . Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Polar coordinate system."CC BYhttp://en.wikipedia.org/wiki/Polar_coordinate_system%23ConventionsView on Boundless.com

  29. Differential Equations, Parametric Equations, a... Solution to the equation The solutions to the equations are periodic. The predator population follows the prey population. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Lotka–Volterra equation."CC BYhttp://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equationView on Boundless.com

  30. Differential Equations, Parametric Equations, a... Exponential Function as a Power Series The exponential function (in blue), and the sum of the first terms of its Maclaurin power series (in red). Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Power series."CC BYhttp://en.wikipedia.org/wiki/Power_seriesView on Boundless.com

  31. Differential Equations, Parametric Equations, a... A Convergent Sequence The plot of a convergent sequence () is shown in blue. Visually, we can see that the sequence is converging to the limit of as increases. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Sequence."CC BYhttp://en.wikipedia.org/wiki/SequenceView on Boundless.com

  32. Differential Equations, Parametric Equations, a... Geometric Sum Visualization of the geometric sum in Example 2. The length of the line () can contain all the successive segments marked off of lengths , , , etc. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Zeno's paradoxes."CC BYhttp://en.wikipedia.org/wiki/Zeno's_paradoxesView on Boundless.com

  33. Differential Equations, Parametric Equations, a... Approximating Arc Length with Hypotenuses The length of a curve can be approximated by using a series of right triangles with the hypotenuses lying along the curve. The smaller the triangles one uses, the closer the approximation will be. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Arc length."CC BYhttp://en.wikipedia.org/wiki/Arc_lengthView on Boundless.com

  34. Differential Equations, Parametric Equations, a... Ratio Test In this example, the ratio of adjacent terms in the blue sequence converges to . We choose . Then the blue sequence is dominated by the red sequence for all . The red sequence converges, so the blue sequence does as well. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Ratio test proof."CC BY-SAhttp://en.wikipedia.org/wiki/File:Ratio_test_proof.svgView on Boundless.com

  35. Differential Equations, Parametric Equations, a... Visual Model of Heat Transfer Visualization of heat transfer in a pump casing, created by solving the heat equation. Heat is being generated internally in the casing and being cooled at the boundary, providing a steady state temperature distribution. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Differential equation."CC BYhttp://en.wikipedia.org/wiki/Differential_equation%23Notable_differential_equationsView on Boundless.com

  36. Differential Equations, Parametric Equations, a... Parametric Example One example of a sketch defined by parametric equations. Note that it is graphed on parametric axes. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Parametric equations."CC BYhttp://en.wikipedia.org/wiki/Parametric_equations%23UsefulnessView on Boundless.com

  37. Differential Equations, Parametric Equations, a... Integral Test The integral test applied to the harmonic series. Since the area under the curve for is infinite, the total area of the rectangles must be infinite as well. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Integral test for convergence."CC BYhttp://en.wikipedia.org/wiki/Integral_test_for_convergenceView on Boundless.com

  38. Differential Equations, Parametric Equations, a... Taylor Polynomials As more terms are added to the Taylor polynomial, it approaches the correct function. This image shows  and its Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Taylor series."CC BYhttp://en.wikipedia.org/wiki/Taylor_seriesView on Boundless.com

  39. Differential Equations, Parametric Equations, a... Limit Convergence Test The ratio between and for is . Since the sum of the sequence diverges, the limit convergence test tells that the original series (with ) also diverges. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Sequence."CC BYhttp://en.wikipedia.org/wiki/SequenceView on Boundless.com

  40. Differential Equations, Parametric Equations, a... Non-Relativistic Schrödinger Equation A wave function which satisfies the non-relativistic Schrödinger equation with . This corresponds to a particle traveling freely through empty space. The real part of the wave function is plotted here. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikimedia.CC BY-SAhttp://upload.wikimedia.org/wikipedia/commons/b/b0/Wave_packet_(dispersion).gifView on Boundless.com

  41. Differential Equations, Parametric Equations, a... Attribution • Wikipedia."derivative."CC BY-SA 3.0http://en.wikipedia.org/wiki/derivative • Wikipedia."Differential equation."CC BY-SA 3.0http://en.wikipedia.org/wiki/Differential_equation • Wiktionary."function."CC BY-SA 3.0http://en.wiktionary.org/wiki/function • Wikipedia."Differential equation."CC BY-SA 3.0http://en.wikipedia.org/wiki/Differential_equation%23Notable_differential_equations • Wikipedia."Radioactive decay."CC BY-SA 3.0http://en.wikipedia.org/wiki/Radioactive_decay • Wiktionary."decay."CC BY-SA 3.0http://en.wiktionary.org/wiki/decay • Wiktionary."differential equation."CC BY-SA 3.0http://en.wiktionary.org/wiki/differential_equation • Wikipedia."Slope field."CC BY-SA 3.0http://en.wikipedia.org/wiki/Slope_field • Wikipedia."Euler method."CC BY-SA 3.0http://en.wikipedia.org/wiki/Euler_method • Wiktionary."differential equation."CC BY-SA 3.0http://en.wiktionary.org/wiki/differential_equation • Wiktionary."normalize."CC BY-SA 3.0http://en.wiktionary.org/wiki/normalize • Wiktionary."tangent."CC BY-SA 3.0http://en.wiktionary.org/wiki/tangent • Wikipedia."derivative."CC BY-SA 3.0http://en.wikipedia.org/wiki/derivative • Wikipedia."Separable partial differential equation."CC BY-SA 3.0http://en.wikipedia.org/wiki/Separable_partial_differential_equation • Wiktionary."differential equation."CC BY-SA 3.0http://en.wiktionary.org/wiki/differential_equation • Wiktionary."fraction."CC BY-SA 3.0http://en.wiktionary.org/wiki/fraction • Wikipedia."derivative."CC BY-SA 3.0http://en.wikipedia.org/wiki/derivative Free to share, print, make copies and changes. Get yours at www.boundless.com

  42. Differential Equations, Parametric Equations, a... • Wikipedia."Logistic function."CC BY-SA 3.0http://en.wikipedia.org/wiki/Logistic_function • Wikipedia."non-linear differential equation."CC BY-SA 3.0http://en.wikipedia.org/wiki/non-linear%20differential%20equation • Wikipedia."boundary condition."CC BY-SA 3.0http://en.wikipedia.org/wiki/boundary%20condition • Wikipedia."Linear differential equation."CC BY-SA 3.0http://en.wikipedia.org/wiki/Linear_differential_equation • Wikipedia."Linear equations."CC BY-SA 3.0http://en.wikipedia.org/wiki/Linear_equations • Wikipedia."simultaneous equations."CC BY-SA 3.0http://en.wikipedia.org/wiki/simultaneous%20equations • Wiktionary."linear equation."CC BY-SA 3.0http://en.wiktionary.org/wiki/linear_equation • Wiktionary."differential equation."CC BY-SA 3.0http://en.wiktionary.org/wiki/differential_equation • Wikipedia."Lotka–Volterra equation."CC BY-SA 3.0http://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equation • Wiktionary."prey."CC BY-SA 3.0http://en.wiktionary.org/wiki/prey • Wiktionary."predator."CC BY-SA 3.0http://en.wiktionary.org/wiki/predator • Wiktionary."differential equation."CC BY-SA 3.0http://en.wiktionary.org/wiki/differential_equation • Wiktionary."coordinate."CC BY-SA 3.0http://en.wiktionary.org/wiki/coordinate • Wikipedia."Parametric equations."CC BY-SA 3.0http://en.wikipedia.org/wiki/Parametric_equations • Wikipedia."Parametric equation."CC BY-SA 3.0http://en.wikipedia.org/wiki/Parametric_equation • Wiktionary."trajectory."CC BY-SA 3.0http://en.wiktionary.org/wiki/trajectory • Wiktionary."displacement."CC BY-SA 3.0http://en.wiktionary.org/wiki/displacement • Wiktionary."acceleration."CC BY-SA 3.0http://en.wiktionary.org/wiki/acceleration • Wikipedia."Polar coordinate system."CC BY-SA 3.0http://en.wikipedia.org/wiki/Polar_coordinate_system Free to share, print, make copies and changes. Get yours at www.boundless.com

  43. Differential Equations, Parametric Equations, a... • Wiktionary."polar."CC BY-SA 3.0http://en.wiktionary.org/wiki/polar • Wikipedia."Polar coordinate system."CC BY-SA 3.0http://en.wikipedia.org/wiki/Polar_coordinate_system • Wikipedia."Polar coordinate system."CC BY-SA 3.0http://en.wikipedia.org/wiki/Polar_coordinate_system • Wiktionary."polar."CC BY-SA 3.0http://en.wiktionary.org/wiki/polar • Wikipedia."Conic section."CC BY-SA 3.0http://en.wikipedia.org/wiki/Conic_section • Wiktionary."cone."CC BY-SA 3.0http://en.wiktionary.org/wiki/cone • Wikipedia."Conic section."CC BY-SA 3.0http://en.wikipedia.org/wiki/Conic_section%23Polar_coordinates • Wiktionary."hyperbola."CC BY-SA 3.0http://en.wiktionary.org/wiki/hyperbola • Wiktionary."cone."CC BY-SA 3.0http://en.wiktionary.org/wiki/cone • Wikipedia."Arc length."CC BY-SA 3.0http://en.wikipedia.org/wiki/Arc_length • Boundless Learning."Boundless."CC BY-SA 3.0http://www.boundless.com//algebra/definition/pythagorean-theorem • Wiktionary."curve."CC BY-SA 3.0http://en.wiktionary.org/wiki/curve • Wikipedia."Sequence."CC BY-SA 3.0http://en.wikipedia.org/wiki/Sequence • Wiktionary."recursion."CC BY-SA 3.0http://en.wiktionary.org/wiki/recursion • Wiktionary."set."CC BY-SA 3.0http://en.wiktionary.org/wiki/set • Wikipedia."Zeno's dichotomy."CC BY-SA 3.0http://en.wikipedia.org/wiki/Zeno's%20dichotomy • Wikipedia."Series (mathematics)."CC BY-SA 3.0http://en.wikipedia.org/wiki/Series_(mathematics) • Wiktionary."sequence."CC BY-SA 3.0http://en.wiktionary.org/wiki/sequence • Wikipedia."Integral test for convergence."CC BY-SA 3.0http://en.wikipedia.org/wiki/Integral_test_for_convergence Free to share, print, make copies and changes. Get yours at www.boundless.com

  44. Differential Equations, Parametric Equations, a... • Wiktionary."improper integral."CC BY-SA 3.0http://en.wiktionary.org/wiki/improper_integral • Wiktionary."natural logarithm."CC BY-SA 3.0http://en.wiktionary.org/wiki/natural_logarithm • Wikipedia."Limit comparison test."CC BY-SA 3.0http://en.wikipedia.org/wiki/Limit_comparison_test • Wikipedia."Direct comparison test."CC BY-SA 3.0http://en.wikipedia.org/wiki/Direct_comparison_test • Wiktionary."improper integral."CC BY-SA 3.0http://en.wiktionary.org/wiki/improper_integral • Boundless Learning."Boundless."CC BY-SA 3.0http://www.boundless.com//calculus/definition/integral-test • Wikipedia."Alternating series."CC BY-SA 3.0http://en.wikipedia.org/wiki/Alternating_series • Wikipedia."Cauchy sequence."CC BY-SA 3.0http://en.wikipedia.org/wiki/Cauchy%20sequence • Wiktionary."monotone."CC BY-SA 3.0http://en.wiktionary.org/wiki/monotone • Wikipedia."Ratio test."CC BY-SA 3.0http://en.wikipedia.org/wiki/Ratio_test • Wikipedia."Root test."CC BY-SA 3.0http://en.wikipedia.org/wiki/Root_test • Wikipedia."Absolute convergence."CC BY-SA 3.0http://en.wikipedia.org/wiki/Absolute_convergence • Wiktionary."summand."CC BY-SA 3.0http://en.wiktionary.org/wiki/summand • Wiktionary."improper integral."CC BY-SA 3.0http://en.wiktionary.org/wiki/improper_integral • Wiktionary."limit superior."CC BY-SA 3.0http://en.wiktionary.org/wiki/limit_superior • Wikipedia."conditional convergence."CC BY-SA 3.0http://en.wikipedia.org/wiki/conditional%20convergence • Wikipedia."Convergence tests."CC BY-SA 3.0http://en.wikipedia.org/wiki/Convergence_tests • Wikipedia."Power series."CC BY-SA 3.0http://en.wikipedia.org/wiki/Power_series • Wiktionary."Z-transform."CC BY-SA 3.0http://en.wiktionary.org/wiki/Z-transform Free to share, print, make copies and changes. Get yours at www.boundless.com

  45. Differential Equations, Parametric Equations, a... • Wiktionary."combinatorics."CC BY-SA 3.0http://en.wiktionary.org/wiki/combinatorics • Wikipedia."Power function."CC BY-SA 3.0http://en.wikipedia.org/wiki/Power_function • Wikipedia."Taylor series."CC BY-SA 3.0http://en.wikipedia.org/wiki/Taylor_series • Wiktionary."differentiable."CC BY-SA 3.0http://en.wiktionary.org/wiki/differentiable • Wiktionary."power law."CC BY-SA 3.0http://en.wiktionary.org/wiki/power_law • Wikipedia."Taylor series."CC BY-SA 3.0http://en.wikipedia.org/wiki/Taylor_series • Wiktionary."analytic function."CC BY-SA 3.0http://en.wiktionary.org/wiki/analytic_function • Wiktionary."differentiable."CC BY-SA 3.0http://en.wiktionary.org/wiki/differentiable • Wikipedia."Taylor series."CC BY-SA 3.0http://en.wikipedia.org/wiki/Taylor_series • Wikipedia."complex analysis."CC BY-SA 3.0http://en.wikipedia.org/wiki/complex%20analysis • Wiktionary."definite integral."CC BY-SA 3.0http://en.wiktionary.org/wiki/definite_integral • Wiktionary."analytic function."CC BY-SA 3.0http://en.wiktionary.org/wiki/analytic_function • Wikipedia."Series (mathematics)."CC BY-SA 3.0http://en.wikipedia.org/wiki/Series_(mathematics)%23Non-negative_terms • Wikipedia."Infinite series."CC BY-SA 3.0http://en.wikipedia.org/wiki/Infinite_series • Wiktionary."limit."CC BY-SA 3.0http://en.wiktionary.org/wiki/limit • Wiktionary."sequence."CC BY-SA 3.0http://en.wiktionary.org/wiki/sequence • Wikipedia."Series (mathematics)."CC BY-SA 3.0http://en.wikipedia.org/wiki/Series_(mathematics)%23Non-negative_terms • Wikipedia."convergence test."CC BY-SA 3.0http://en.wikipedia.org/wiki/convergence%20test • Wiktionary."converge."CC BY-SA 3.0http://en.wiktionary.org/wiki/converge Free to share, print, make copies and changes. Get yours at www.boundless.com

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