# Series Convergence

Please see attached file for full problem description.

1) Consider the series where . Show that and for .

2) Use the result of the previous problem to find .

3) The series converges. Find its sum.

4) Determine whether the series converges or diverges. Fully justify your answer.

5) Determine whether the series converges or diverges. Fully justify your answer.

6) Determine whether the series converges or diverges. Fully justify your answer.

7) The series converges. Find an upper bound for the sum. Fully justify your answer.

8) Determine all values of x for which the power series converges.

9) Show that if . Using this, find a power series for centered at 0.

10) If then . Find the Taylor Series for centered at .

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#### Solution Preview

• Please provide VERY CLEAR AND DETAILED SOLUTIONS.

• Please provide exact answers, not decimal approximations.

1) Consider the series where . Show that and for .

We have . Plugging the value n=1, we get the first term of the series i.e.,

We have and . The k th term is nothing but the difference between the sum up to k terms and the sum up to (k-1) terms.

So

.

Hence we can write for .

2) Use the result of the previous problem to find .

Here

We can write as . We know that sum of the series as from the previous result.

Hence =

3) The series converges. Find its ...

#### Solution Summary

Series convergence and divergence are investigated.