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Power Series. Dr. Dillon Calculus II Fall 1999. Recall the Taylor polynomial of degree. for a function. times differentiable at. which is. Taylor Polynomials. The Taylor series for a function. which has derivatives of all orders at a point. Definition. is given by. Compare.

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Power series

Power Series

Dr. Dillon

Calculus II

Fall 1999


Taylor polynomials

Recall the Taylor polynomial of degree

for a function

times differentiable at

which is

Taylor Polynomials


Definition

The Taylor series for a function

which has derivatives of all orders at a point

Definition

is given by


Compare
Compare

Taylor polynomial

Taylor series


A taylor polynomial
A Taylor Polynomial

is a (finite) sum

with a defined degree.


A taylor series
A Taylor Series

is an infinite sum, i.e.,

a sequence of partial sums.


Sequence of partial sums
Sequence of Partial Sums

Each member of the sequence of

partial sums is itself a Taylor

polynomial.

A Taylor series is a

sequence of Taylor polynomials.


Sequence of taylor polynomials

Let

Sequence of Taylor Polynomials


As the taylor series

at

to be the sequence

As the Taylor Series

We actually define the Taylor series for


An example

at

An Example

The first few Taylor polynomials for

The Taylor series:


Assumptions

you need derivatives up to order

Assumptions

For a Taylor polynomial,

For a Taylor series,

you need derivatives of all orders.


Examples

at

at

at

at

Examples

We could find Taylor series for


Notation

at

for the Taylor series for

Notation

Using sigma notation write


Terminology

at

The Taylor series for

is called the Maclaurin series for

Terminology


Will the calculator find a taylor series

at

Will the Calculator Find a Taylor Series?

Maybe; try it for


Where does one start
Where Does One Start?

Start with a Taylor polynomial.

Try using a large degree.

Can you see the pattern?

I can’t!


Now what

Find the coefficient of the degree

term

at

derivative of

Start by finding the

Now What?

in the Taylor series for


Use the calculator to
Use the Calculator to...

get successive expressions for


The pattern
The Pattern?

Look at a couple more:

...


Conclusion

is good for

and then

Conclusion



The coefficient for the degree

term is

at

Thus

in the Taylor series for


Notice
Notice

Finding a Taylor series

means finding the coefficients.


Taylor series
Taylor Series

Degree n term

Coefficient of Degree n Term


Separate issues
Separate Issues

  • What is the Taylor series for a function at a point?

  • For what values of x does the Taylor series for a function converge?

  • Does the Taylor series for a function converge to that function?


A weird example

Let

are all

at

A Weird Example

It’s hard to see that it even exists, but

Thus the coefficients in the Taylor series for


Meaning
Meaning...

This Taylor series describes the function well

but only at one point, 0.

In cases like this, Taylor series aren’t good for much.


Lucky for us
Lucky for Us

All of the usual suspects can be well represented by their Taylor series at all points where they are infinitely differentiable.

The Taylor series for all of our favorite functions converge to the functions at least on a decent sized interval, if not on the entire real line.


Meaning1
Meaning...

  • For all algebraic functions,

  • for trigonometric functions and their inverses,

  • for exponential functions and logarithms,

    there are excellent polynomial approximations to the functions on intervals surrounding points

where the functions have derivatives

of all orders.


Example

that is to within two decimal places of

on the interval (2,6).

Use a Taylor polynomial at

Example

Find a polynomial approximation to

Solution?


What degree do we need
What Degree Do We Need?

Use your technology to figure it out!


How can you tell
How Can You Tell?

If the polynomial and the function

agree at the endpoints,

they agree at all the points between.

That’s a Big Theorem.


Power series1

A power series at

is a function of the form

The

Power Series

are the coefficients.


Example1

is a power series at

Example

All the coefficients are 1.



If

for some positive number

then

Huge Theorem

is the Taylor series for f at a.


So...

is the Maclaurin series for


More interesting stuff
More Interesting Stuff

If a power series converges on an interval, we can

  • differentiate term by term to get another convergent power series

  • integrate term by term to get another convergent power series

  • take limits term by term, on the interval of convergence

  • do arithmetic term by term to get still more convergent power series.


Example2

converges to

Example

In other words,

for a given value of x


True for all x

for

means we can substitute

True For All x...

That gives us

which is the Maclaurin series for that function.



One last example

Find the Maclaurin series for

Start with the Maclaurin series for

It converges to

One Last Example

Solution


This is the Maclaurin series for

Thus...

It converges to the function on the whole real line.


A final note
A Final Note

Memorize the following series:

  • The Maclaurin series for sine and cosine

  • The Maclaurin series for the natural exponential

The Maclaurin series for 1/(1-x)


Be able to use them
Be Able to Use Them

to find Taylor series for

functions obtained from the above via

  • Differentiation

  • Integration

Arithmetic


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