# Solutions to Problem Set 26

Math 211-03

11-9-2017

[Constructing power series]

1. Construct the power series centered at for . Write your answer in summation form, or give the first 4 nonzero terms of the series. In addition, find the interval of convergence.

Let in the series for .

For the interval of convergence, I have . Since a square can't be negative, the relevant part of the inequality is . This gives , or .

2. Construct the power series centered at for . Write your answer in summation form, or give the first 4 nonzero terms of the series. In addition, find the interval of convergence.

Letting in the series for , I get

For the interval of convergence,

3. Construct the power series centered at for . Write your answer in summation form, or give the first 4 nonzero terms of the series. In addition, find the interval of convergence.

Letting in the series for , I get

For the interval of convergence,

4. Construct the power series centered at for . Write your answer in summation form, or give the first 4 nonzero terms of the series. In addition, find the interval of convergence.

Letting in the series for , I get

For the interval of convergence,

5. Construct the power series centered at for . Write your answer in summation form, or give the first 4 nonzero terms of the series. In addition, find the interval of convergence.

Letting in the series for , I get

For the interval of convergence,

6. Construct the power series centered at for . Write your answer in summation form, or give the first 4 nonzero terms of the series. In addition, find the interval of convergence.

Letting in the series for , I get

For the interval of convergence,

7. Construct the power series centered at for . Write your answer in summation form, or give the first 4 nonzero terms of the series. In addition, find the interval of convergence.

The series is centered at , so I want powers of .

Letting in the series for , I get

For the interval of convergence,

8. Find the function to which the power series converges.

Letting in the series for , I find that

9. Find the function to which the power series converges.

Letting in the series for , I find that

Love is not consolation, it is light. - Simone Weil

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