Proof that $E[X^2]$ = $\sum_{n=1}^\infty (2n-1) P(X\ge n)$

X is a random variable with values from $ \Bbb N\setminus{0}$

I am trying to show that $ E[X^2]$ = $ \sum_{n=1}^\infty (2n-1) P(X\ge n)$ iff $ E[X^2]$ < $ \infty$ .

I rewrote $ P(X \ge n)$ :

$ E[X^2]$ = $ \sum_{n=1}^\infty (2n-1)\sum_{x=1}^\infty 1_{x \ge n}P(X=x)$

Now I tried to rearrange the sums:

$ E[X^2]$ = $ \sum_{x=1}^\infty \sum_{n=1}^x (2n-1)P(X=x)$

But I think that I made a mistake. Could you give me some hints?