How to compute an upper bound of $1 + \frac{1}{2^a} + \frac{1}{3^a} + \cdots$? Suppose $a>2$.
How to compute an upper bound of $1 + \frac{1}{2^a} + \frac{1}{3^a} + \cdots$ ? 
Is $\frac{1}{a-2}$ an upper bound?

When a = 2 we can say
$1 + \frac 14 (1 + \frac {4}{9}) + \frac {1}{16}( 1 + \frac {16}{25} + \frac {16}{36} + \frac {16}{49}) + \cdots < 1 + \frac 14 (2) + \frac 1{16} (4) + \cdots = 2$
when $a> 1,$
$\sum_\limits{n=0}^{\infty} \frac {1}{n^a} < \sum_\limits{n=0}^{\infty} (\frac {1}{2^{a-1}})^n  = \frac {2^{a-1}}{2^{a-1}-1}$
