# Solutions to Problem Set 27

Math 211-03

11-13-2017

[Taylor's formula]

1. Suppose that

Use the degree Taylor polynomial to approximate .

2. Suppose that

Use the degree Taylor polynomial to approximate .

3. Suppose that , and for ,

Write down the degree Taylor polynomial (with each of the coefficients simplified).

So

4. The degree term in the Taylor series for at is . Find .

By Taylor's formula, the degree term in the Taylor series for at is .

So

5. Find the degree Taylor polynomial for expanded at .

So

Therefore,

6. Find the degree Taylor polynomial for expanded at .

Then

Hence,

[Taylor series applications]

7. (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)

(b) Using the series in (a),

Hence,

8. (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) Set in the series for :

(b) Using the series in (a),

Hence,

Too many people overvalue what they are not and undervalue what they are. - Malcolm Forbes

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