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!