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Section 9.2 – Series and Convergence

Section 9.2 – Series and Convergence. Goals of Chapter 9. Approximate Pi Prove infinite series are another important application of limits, derivatives, approximation, slope, and concavity of functions . Find challenging antiderivatives like Lay the groundwork for future courses.

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Section 9.2 – Series and Convergence

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  1. Section 9.2 – Series and Convergence

  2. Goals of Chapter 9 • Approximate Pi • Prove infinite series are another important application of limits, derivatives, approximation, slope, and concavity of functions. • Find challenging antiderivatives like • Lay the groundwork for future courses

  3. Summation Notation A compact notation (often called sigma notation) for sums is the following: Upper Limit of Summation General Term Index of Summation Lower Limit of Summation

  4. Examples Evaluate: i=3 i=1 i=2 Series investigate the following:

  5. Infinite Sum 1 What is the area of the square? square unit Cut the square in half and label the area of one section. Cut the unlabeled area in half and label the area of one section. 1 Continue the process… Since the infinite sum represents the area of the square… Sum all of the areas: The general term is…

  6. Infinite Series An infinite series is an expression of the form , or The numbers are the terms of the series; is the nth term.

  7. Connecting Series and Sequences Consider the Series: Find the sum of… The first term: The first 2 terms: The first 3 terms: The first 4 terms: Consider the sequence of PARTIAL SUMS above: The sequence of PARTIAL SUMS appear to converge to:

  8. Partial Sums of a Series The partial sums of the series for a sequence: of real numbers, each defined as a finite sum.

  9. Convergent or Divergent Series If the sequence of partial sums has a limit as , we say the series converges to the sum , and we write Otherwise, we say the series diverges.

  10. Examples Investigate the partial sums of the sequences below to determine if the series converges or diverges. If it converges, state the limit. The limit of the general term does not equal 0. Why do 1, 3, 4 and 6 Diverge?

  11. The n-th Term Test If , then the infinite seriesdiverges. OR If the infinite series converges, then . When determining if a series converges, always use this test first! Is the converse of this statement true? If , does the infinite series always converge?

  12. The Converse of The n-th Term Test Consider the two famous sequences below: For both series’, the .BUT do both series’ converge? Check a calculator program.

  13. The Converse of The n-th Term Test The Alternating Harmonic Series appears to converge to ~0.69. The Harmonic Series appears to diverge.

  14. The n-th Term Test If , then the infinite seriesdiverges. OR If the infinite series converges, then . When determining if a series converges, always use this test first! The converse of this statement is NOT true. If , the infinite series does not necessarily converge.

  15. The Harmonic Series Diverges Prove the Harmonic Series diverges: Compare the Series to the graph of . Find the Left Hand Riemann Sum to approximate . The Left Hand Riemann Sum is equal to the Sum of the Harmonic Series. … …

  16. The Harmonic Series Diverges Prove the Harmonic Series diverges: Compare the Series to the graph of . So… Since is decreasing, the Left Hand Riemann Sum is an over estimate. Thus: …

  17. The Harmonic Series Diverges Prove the Harmonic Series diverges: So… Compare the Series to the graph of . We can find the value of the improper integral: Since diverges and , the Harmonic Series Diverges. …

  18. The Harmonic Series Diverges Part 2 Justify that the Harmonic Series diverges another way: Investigate the sum: By increasing the size of , we can make the sum of the infinite series as large was we desire.

  19. The Alternating Harmonic Series Converges Justify the Alternating Harmonic Series converges: Investigate and plot the sum: We will find the actual value of the sum soon. Each Successive term in the sequence of partial sums is between the two previous terms in this sequence . The sum is bounded by 0.5 and 1. The sum must be between any two successive terms.

  20. Arithmetic and Geometric Series An Arithmetic Series has a constant difference between terms. (Similar to an Arithmetic Sequence.) Example: A Geometric Series has a constant ratio between terms. (Similar to a Geometric Sequence.) Example:

  21. Arithmetic and Geometric Series By the n-th Term Test, every Arithmetic Series diverges: Some Geometric Series diverge and others converge: Since Geometric Series occasionally converge, we will focus on them.

  22. Definition of a Geometric Series In a geometric series each term is obtained from its preceding term by multiplying by the same number : Examples: The previous examples are geometric.

  23. White Board Challenge Find the general term and the sum of the first 10 terms of the sequence:

  24. Finite Sum of a Geometric Series Find the sum of the first terms of a geometric series: Multiply by r. Subtract the two equations. Solve for the sum. What happens to the sum as the value of n increases to infinity? Check with the previous example.

  25. Infinite Sum of a Geometric Series Depends on the value of r. Consider :

  26. Convergent Geometric Series The geometric series converges if and only if . If the series converges, its sum is . Example: Find the sum if it exists. Where a is the first term and r is the constant ratio.

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