# Solutions to Problem Set 23

Math 211-03

11-2-2017

[Alternating series]

1. Determine whether the series converges or diverges.

The terms alternate, and

If , then

Hence, the terms decrease in absolute value.

Therefore, the series converges by the Alternating Series Test.

2. Determine whether the series converges or diverges.

The terms alternate, and

If , then

Hence, the terms decrease in absolute value.

Therefore, the series converges by the Alternating Series Test.

3. Determine whether the series converges or diverges.

The terms alternate, but

The series diverges by the Zero Limit Test.

4. Determine whether the series converges or diverges.

The terms alternate, and

If , then

Now for , so for . Hence, the terms decrease in absolute value.

The series converges by the Alternating Series Test.

5. Determine whether the series converges or diverges.

The terms alternate, and by L'H\^opital's rule

Let . Then

Hence, the terms decrease in absolute value.

Therefore, the series converges by the Alternating Series Test.

6. Determine whether the series converges or diverges.

The terms alternate. Using L'H\^opital's Rule, I have

Hence,

Therefore, the series diverges by the Zero Limit Test.

7. Consider the convergent alternating series .

Find the smallest value of n for which the partial sum approximates the actual sum to within 0.1.

The partial sum differs from the actual sum s by no more than the absolute value of the next term :

I can ensure that if . Then

So

Thus, .

8. Consider the convergent alternating series .

Find the smallest value of n for which the partial sum approximates the actual sum to within 0.001.

The partial sum differs from the actual sum s by no more than the absolute value of the next term :

I can ensure that if . Then

That is,

However, I can't solve this inequality algebraically. Therefore, I'll do this by trial and error by making a table. (I'm only showing some of the values.)

I see that for the first time when . So I need to use to estimate the sum to within 0.001.

Without work, all life goes rotten. But when work is soulless, life stifles and dies. - Albert Camus

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