Approximation of the basel problem

Question

Why is it, that $$\sum_{n=1}^{x} \frac1{n^2} $$ is about equal to $(1+\frac1{2x})^{x-1}$? Is it possible to give an error term? My observation is, that for $x=1,2,3$ the two terms are equal but with $x=4$, there is a difference of $0.004$. Any hints or solutions are appreciatet.