let ${x_n}$ be the sequence of the positive roots of the equation $tan (x)=x$

Prove/disprove: the series $∑1/{x^2_n}$ converges.

I spent a lot of time trying to prove it, but I do not think I'm doing well.

By looking at the graph, I know that for any natural n, $nπ<x_n<(n+1/2)π$

So let $a_n=1/((n+1/2)π)^2$ and $c_n=1/(nπ)^2$$

Using the comparison test with the convergent series $ ∑1/{n^2}$ and $p_n = 1/{n^2}$ we have:

$\lim_{n\rightarrow∞} (a_n/p_n) = \lim_{n\rightarrow∞} (1/((n+1/2)π)^2 )/(1/n^2 ) = 1/\pi^2$ $\lim_{n\rightarrow∞} (c_n/p_n) = \lim_{n\rightarrow∞} (1/(nπ)^2 )/(1/n^2 ) = 1/\pi^2$

So, $ ∑1/{a_n}$ and $ ∑1/{c_n}$ are convergent.

let $ε>0$.

There is some N such that for any $n,k>N$ , $ |∑1/{a_n}| < ε$ and $ |∑1/{c_n}|< ε$

We know that $a_n≤1/(x_n^2 )≤c_n$, so $ ∑1/{a_n}≤∑1/{x^2_n}≤∑1/{c_n}$

Therefore $-ε<∑1/{x^2_n}<ε$

Which leading the conclusion: according to Cauchy convergence test, the series $∑1/{x^2_n}$ convergent.

Questions:

The graph conclusions are correct?

How can I prove mathematically what I assumed from the graph?

Is the rest of the proof correct?

Can someone prove or disprove the problem from zero in a formal way? I have been reading every bit of information on Stack and the Internet, I couldnt find a full proof to this problem.