# Solutions to Problem Set 28

Math 211-03

11-14-2017

[Taylor series applications]

1. (a) Find the first 4 nonzero terms of the Taylor series at for .

(b) Use the series in (a) to obtain the value of the limit .

(a) Use

Letting , I get

(b) Using the series in (a),

Hence,

2. (a) Find the first 4 nonzero terms of the Taylor series for at .

(b) Use the 4 terms of the series you found in (a) to find the first 4 nonzero terms of the series for at .

(a) Use

Let . This gives

(b) Using the first four terms, I get

3. (a) Assume that the following function is defined and differentiable at :

Find the first 4 nonzero terms of the Taylor series for at .

(b) Use the 4 terms of the series you found in (a) to approximate .

(a)

(b) Using the first four terms, I get

[Differentiating and integrating series]

4. (a) Use a known series to construct the power series centered at for . Write your answer in summation form, or give the first four nonzero terms of the series.

(b) By differentiating the series in (a), construct the power series centered at for . Write your answer in summation form, or give the first four nonzero terms of the series.

(a) I have

Set :

(b) Note that

Hence,

5. (a) Use a known series to construct the power series centered at for . Write your answer in summation form, or give the first four nonzero terms of the series.

(b) By differentiating the series in (a), construct the power series centered at for . Write your answer in summation form, or give the first four nonzero terms of the series.

(a) I have

Set :

(b) I have

So

6. (a) Write down the first 3 nonzero terms of the binomial series expansion for at .

(b) By integrating the series in (a) from to , find the first three nonzero terms of the power series centered at for .

(a) I have

Set and :

(b) I have

7. (a) Use a known series to construct the power series centered at for . Write your answer in summation form, or give the first four nonzero terms of the series.

(b) By integrating the series in (a) from to , find the power series centered at for . Write your answer in summation form, or give the first four nonzero terms of the series.

(a) I have

Set :

(b) I have

In summation form,

Courage consists of the power of self-recovery. - Ralph Waldo Emerson

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