I’ve always been boggled by this:

Summation from n = 1 to (infinity) of : 1/n^p

= 1/1^p + 1/2^p + 1/3^p + 1/4^p + 1/5^p +… + 1/n^p + …

This series is said to converge if p > 1, but it diverges if p <= 1

at p = 1, look what’s going on here:

1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 … + 1/n + …

Now the good ol’ intuition starts working here and i’m thinking, i’m adding on successively smaller and smaller numbers. I’m thinking *convergence*, but sure as $#!t the S.O.B. goes and fails the damn integral test:

Summation from n = 1 to (infinity) of 1/n =

Integral from N to (infinity) (f(x) dx):

So I work it out:

Integral from 1 to (infinity) (1/x dx) =

limit as k → (infinity) Integral from 1 to k (1/x dx) =

limit as k → (infinity) ln(x) evaluated at x = k and subtract ln(x) evaluated at x = 1

= limit as k → (infinity) ln(k) - ln(1)

I want to get that ln out of there so i raise E to both sides of the equation and get:

limit as k → (infinity) k - 1

DAMN! that sucker pretty much equals (infinity) - 1; which is still infinity. Has anyone else found this harmonic series convergence/divergence stuff baffling?? and AND why is it that p = 1 seems to be the magic number, any number greater than one ( even 1.000000000000000000000000001) would cause this series to converge and any number less than 1 (even 0.9999999999999999999999999999) diverges.

Thoughts anyone???