How can I find the number equaling 2 in Riemann's zeta function?

Question

I used some graphs but did not find the correct value,

$$\sum_{n=1}^ \infty \frac{1}{n^x} = 2 $$

what is the x value?

Answer 1

In order to solve $\zeta(x)=2$ we may use $\zeta(2)=\frac{\pi^2}{6}$ and $\zeta(s)=\frac{1}{s-1}+\gamma+o(1)$ for $s\to 1^+$.
$\zeta(s)$ is decreasing and convex on $(1,2)$, hence Newton's method $$ x \mapsto x-\frac{\zeta(x)-2}{\zeta'(x)} $$ with starting point $x=2$ (or, better, $\frac{3-\gamma}{2-\gamma}$) converges pretty fast to $1.72865\ldots$