* Example.* Expand in a power series at and find the interval of convergence.

For the interval of convergence,

* Example.* Find the interval of convergence of
.

Apply the Root Test:

The series converges for

At , the series is

This series diverges by the Zero Limit Test.

At , the series is

This series also diverges by the Zero Limit Test.

The power series converges for and diverges elsewhere.

* Example.* Expand in a power series at and find the interval of convergence.

Set in

This gives

The interval of convergence for the series is . So for the series,

* Example.* Expand in a Taylor series at .

Using the double angle formula

* Example.* (a) Use the first four nonzero terms
of the Taylor series for at to approximate .

(b) Use the Alternating Series Test to estimate the error in part (a).

(a)

Hence,

(I used the first four terms to get the approximation.)

(b) The error is no greater than the next term, which is .

* Example.* Use the Taylor series expansion of
at to explain the fact
that .

The series for at is

Divide by x to obtain

Then

* Example.* Find the first four nonzero terms of
the Taylor expansion for at .

The series is

* Example.* Find the interval of convergence of
.

The series converges for

At , the series is

This is harmonic, so it diverges.

At , the series is

This is harmonic, so it diverges.

The power series converges for and diverges elsewhere.

* Example.* satisfies

Use the third degree Taylor polynomial for f at to approximate .

The third degree Taylor polynomial for f at is

So

* Example.* Suppose that . Use to estimate the error in using the fourth degree
Taylor polynomial at to approximate for .

For some z between 0 and x,

Since , .

For the z-term, I have . Thus,

So . Therefore,

Copyright 2016 by Bruce Ikenaga