Math Help - Prove inequality?

1. Prove inequality?

If $n$ is a positive integer and $S>1$, prove that:

$\frac{1}{1^S} + \frac{1}{2^S} + \frac{1}{3^S} + \mbox{...} + \frac{1}{n^S} < \frac{1}{1 - 2^{1 - S}}$

2. Originally Posted by fardeen_gen
If $n$ is a positive integer and $S>1$, prove that:

$\frac{1}{1^S} + \frac{1}{2^S} + \frac{1}{3^S} + \mbox{...} + \frac{1}{n^S} < \frac{1}{1 - 2^{1 - S}}$
Hi fardeen_gen.

Observe that

$\frac1{3^s}\ <\ \frac1{2^s}$

$\frac1{5^s}\ <\ \frac1{4^s}$

$\frac1{7^s}\ <\ \frac1{6^s}$

$\vdots$

$\therefore\ 1-\frac1{2^2}+\frac1{3^s}-\frac1{4^s}+\cdots+\frac1{(2k)^s}\ <\ 1$

$\implies\ \sum_{n\,=\,1}^\infty\frac{(-1)^{n+1}}{n^s}\ <\ 1$

$\therefore\ \frac1{1^s}+\frac1{2^2}+\cdots+\frac1{n^s}\ <\ \zeta(s)\ =\ \frac1{1-2^{1-s}}\sum_{n\,=\,1}^\infty\frac{(-1)^{n+1}}{n^s}\ <\ \frac1{1-2^{1-s}}$