The problem:

Define .

Determine whether converges or diverges.

My solution:

Define

.

We will use the comparison test to show that since diverges, so too does diverge.

Observe that if , we may divide both sides by to yield

.

Adding to both sides gives us

.

Dividing by , we have

.

Since and when is odd, then we have

when is odd.

Now observe that since , then dividing by gives us

.

Since and when is even, then we have

when is even.

Thus, whether is odd or even, we have , with divergent. By the comparison test, therefore is likewise divergent.

The textbook tells me I am wrong, and that this series is actually convergent.

Any ideas what went wrong, here?

Thanks!