True or false series?

Question

With k being all real numbers, and an,bn, and cn being all sequences, Σi=1 1/1^k converges for k>1 and diverges for k< or equal to 1. I know this is true, but I have to prove it. I cannot use Cauchy's test. What is an example/proof of this being true?

Answer 1

Hi Lily S.,

The usual way of showing this is by ratioing the series to itself (using self-similarity), and demonstrating that the sum can be so calculated. So:

let P = sum (n=1 to infinity) of 1/n^k

then k*P = 1 + sum(n=1 to infinity) of 1/n^k (the whole series has been "shifted" by one term)

subtract: P(k-1) = 1

P = 1/(k-1) which exists and is finite for k>1

Right?

-- Cheers, -- Mr. d.