What Value of x Causes the Series Ʃ(1/n^x) to Diverge?

[gottfried]

Ʃ(\frac{1}{n}) diverges but Ʃ(\frac{1}{n^2}) converges but both corresponding sequences converge (I think). So at what rate does a sequence have to converge for the corresponding series to converge.

Or asked differently what is the largest value x can be for Ʃ(\frac{1}{n^x}) to diverge

 

[LCKurtz]

Ʃ(\frac{1}{n}) diverges but Ʃ(\frac{1}{n^2}) converges but both corresponding sequences converge (I think). So at what rate does a sequence have to converge for the corresponding series to converge.

Or asked differently what is the largest value x can be for Ʃ(\frac{1}{n^x}) to diverge

The dividing line is at p = 1. $\sum \frac 1 {x^p}$ converges if $p>1$, as you can verify by the integral test.

 

[gottfried]

Thanks.