Question

There is this series:
1 + (1/2)^p + (1/3)^p + ... + (1/n)^p

Where p is a real number.
- p >1
- p = 1
- p < 1

And my lecturer wants us to proof that the (p<1) series is divergence using comparison test. And he gave us the example of (p>1)

Un -> 1 + (1/2)^p + ... + (1/n)^p
Vn (convergence) -> 1 + 1/2 + 1/4 + ... + 1/(2^(n-1))
remember, p>1
1=1
(1/2)^p + (1/3)^p < (1/2)^p + (1/2)^p
(1/4)^p + ... + (1/7)^p < (1/4)^p + (1/4)^p + (1/4)^p + (1/4)^p
and so on
and because of Un'1) series is convergence and we must proof that in (p<1) Un>Vn