# Solutions to Problem Set 25

Math 211-03

11-7-2017

[Interval of convergence]

1. Find the interval of convergence of the series .

Since the limiting ratio is always less than 1, the series converges for all x. The interval of convergence is .

2. Find the interval of convergence of the series .

The series converges absolutely for . Solving for x, I get

When , the series is

This is the harmonic series, so it diverges.

When , the series is

This is the alternating harmonic series, so it converges.

The interval of convergence is .

3. Find the interval of convergence of the series .

The series converges absolutely for . Solving for x, I get

When , the series is

Since , the series diverges by the Zero Limit Test.

When , the series is

Since is undefined, the series diverges by the Zero Limit Test.

The interval of convergence is .

4. Find the interval of convergence of the series .

Since the limiting ratio is always greater than 1, the series only converges at the expansion point .

5. Find the interval of convergence of the series .

The series converges absolutely for . Solving for x, I get .

When , the series is . This is a p-series with , so it converges.

When , the series is . This is an alternating p-series with , so it converges.

The interval of convergence is .

He who has not lost his head over some things has no head to lose. - Jean-Paul Richter

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