Solutions to Problem Set 18

Math 211-03

10-23-2017

[Integral Test]

1. Determine whether the series converges or diverges.

The series has positive terms.

Let . Then f is continuous for .

Now

Since for , the terms of the series decrease.

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

The integral diverges, so the series diverges, by the Integral Test.

2. Determine whether the series converges or diverges.

The series has positive terms.

Let . Then f is continuous for .

Now

Since for , the terms of the series decrease.

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

The integral converges, so the series converges, by the Integral Test.

3. Determine convergence or divergence of the series .

The series has positive terms.

Let . Then f is continuous for .

Now

for , and for . The series starts at , so the terms of the series decrease.

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

The integral diverges, so the series diverges, by the Integral Test.

4. Determine whether the series converges or diverges.

The series has positive terms.

Let . Then f is continuous for .

Now

If , then , so

Since the top of the fraction for is negative and the bottom is positive, for . Therefore, the terms of the series decrease.

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

The integral converges, so the series converges, by the Integral Test.

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