Help with a subtle point concerning the proof of the p-series test

[JJHK]

Homework Statement

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?

 

[voko]

In the p-series convergence theorem, p is fixed. If it is a bit different from 1, it will stay a bit different from 1. You need not worry about this.