# Solutions to Problem Set 22

Math 211-03

10-30-2017

[Series review]

1. Determine whether the series converges or diverges.

Add the fractions over a common denominator:

Therefore, the series is . Apply Limit Comparison:

The limit is finite and positive. The series converges, because it's a p-series with . Therefore, the original series converges, by Limit Comparison.

2. Determine whether the series converges or diverges.

Apply the Root Test:

The limiting ratio is less than 1, so the series converges, by the Root Test.

3. Determine whether the series converges or diverges.

The terms are positive. Let . Then f is continuous for .

I have

Note that , so when . But x and both increase for . Therefore, for , so the terms of the series decrease.

The conditions for the Integral Test are satisfied. Compute the integral:

Since the integral converges, the series converges by the Integral Test.

4. Determine whether the following series converges or diverges:

The series is . It is a p-series with , so it converges.

5. Determine whether the following series converges or diverges:

Write the series as . I'll use Limit Comparison:

is times the harmonic series, so it diverges. Therefore, the original series diverges by Limit Comparison.

6. Determine whether the series converges or diverges.

Use the Ratio Test:

The limiting ratio is greater than 1, so the series diverges.

7. Determine whether the series converges or diverges.

( and are geometric sequences with ratios and . Since the ratios are less than 1, the sequences go to 0 as n goes to .)

Since does not approach 0 as n goes to , the series diverges by the Zero Limit Test.

8. Use direct comparison to show that the following series converges: .

Changing the top from to makes the top bigger, so the fraction gets bigger. Changing to makes the bottom smaller, so the fraction gets bigger. Together, these changes make the fraction bigger:

The series converges, because it's a p-series with . Therefore, the original series converges by direct comparison.

Nothing is so exhausting as indecision, and nothing is so futile. - Bertrand Russell

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