Yep, but my question is rather how can I find 1.036... without adding terms and without using the zeta function.

For instance it's easy to find $\displaystyle \sum_{n=1}^{\infty} 1/(4n^2+4n)=1/4$ because $\displaystyle \sum_{n=1}^{\infty} 1/(4n^2+4n)= \frac{1}{4} \sum_{n=1}^{\infty} \frac{1}{n} -\frac{1}{n+1}=\frac{1}{4}(1-\frac{1}{2}+\frac{1}{2}- \dots -\frac{1}{n+1})=\frac{1}{4} $ ( \underbrace{}_{} doesn't work

but $\displaystyle \frac{1}{n+1}=0$)

So is there something similar for $\displaystyle \sum_{n = 1}^\infty {\frac{{1}}{{n^5 }}}

$ ?