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Power Series

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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

The Taylor series for a function

which has derivatives of all orders at a point

is given by

Taylor polynomial

Taylor series

is a (finite) sum

with a defined degree.

is an infinite sum, i.e.,

a 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.

Let

at

to be the sequence

We actually define the Taylor series for

at

The first few Taylor polynomials for

The Taylor series:

you need derivatives up to order

For a Taylor polynomial,

For a Taylor series,

you need derivatives of all orders.

at

at

at

at

We could find Taylor series for

at

for the Taylor series for

Using sigma notation write

at

The Taylor series for

is called the Maclaurin series for

at

Maybe; try it for

Start with a Taylor polynomial.

Try using a large degree.

Can you see the pattern?

I can’t!

Find the coefficient of the degree

term

at

derivative of

Start by finding the

in the Taylor series for

get successive expressions for

Look at a couple more:

...

is good for

and then

When

The coefficient for the degree

term is

at

in the Taylor series for

Finding a Taylor series

means finding the coefficients.

Degree n term

Coefficient of Degree n Term

- 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?

Let

are all

at

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

Thus the coefficients in the Taylor series for

This Taylor series describes the function well

but only at one point, 0.

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

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.

- 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.

that is to within two decimal places of

on the interval (2,6).

Use a Taylor polynomial at

Find a polynomial approximation to

Solution?

Use your technology to figure it out!

If the polynomial and the function

agree at the endpoints,

they agree at all the points between.

That’s a Big Theorem.

A power series at

is a function of the form

The

are the coefficients.

is a power series at

All the coefficients are 1.

If

for some positive number

then

Huge Theorem

is the Taylor series for f at a.

is the Maclaurin series for

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.

converges to

In other words,

for a given value of x

for

means we can substitute

That gives us

which is the Maclaurin series for that function.

means

Find the Maclaurin series for

Start with the Maclaurin series for

It converges to

Solution

This is the Maclaurin series for

It converges to the function on the whole real line.

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)

to find Taylor series for

functions obtained from the above via

- Differentiation
- Integration

Arithmetic