Hi, I'm trying to prove the p-series test, using the integral test. Everything seems to work out fine until I get into this point:
Consider when p≠1 (p=1 is easy to see that it diverges, so I will ignore that one).
Then we have:
∫(n)^(-p) dn = [ 1 / (-p+1) ] * n^(-p+1)
For very large (-p+1) or very negatively large (-p+1), it is clear to see that the integral diverges and converges, respectively. But what if (-p+1) → 0? How do we know that the integral converges if p>1 and diverges if p<1? Shouldn't the limit approach 1 as anything to the zeroth power is equal to 1?