Use the Integral Test to determine the convergence or divergence of the series, where k is a positive integer.

The Integral test says that this sum will converge if and only if this integral also converges: When integrating this, we would use integration by parts, and we would need to use it k times. The first set of parts is

The Integral test says that this sum will converge if and only if this integral also converges:

When integrating this, we would use integration by parts, and we would need to use it k times. The first set of parts is

Then we repeat for , and so on until we have only the term left.

But the important thing is that the last term would only be in terms of a constant times , which clearly converges; and then all the other terms would look like this, for some integer
and some constant C:

The value of this term at we can simply calculate; no problem there, it will be some finite number. The limit as x goes to infinity we can also determine by the fact that always increases faster than any polynomial as x gets very large, and thus for any value of p, this limit must be zero.

Thus, we have k-1 terms that are finite (zero minus a finite value), plus one final term that is a convergent integral. Therefore the whole integral converges; therefore the sum converges.