Does the following series converges or diverges? $\sum_{n=1}^{+\infty}\left[n \log\left ( \frac{2n+1}{2n-1} \right )-1\right]$ ,converges or not
My attempt
$a_n=n \log\left ( \frac{2n+1}{2n-1} \right )-1 \\ \\
\log \left ( \frac{2n+1}{2n-1} \right )=\log \left ( 1+\frac{2}{2n-1} \right ) \sim \frac{2}{2n-1} \ (n\rightarrow +\infty) \\ \\
a_n=n\log \left ( 1+\frac{2}{2n-1} \right )-1 \sim n \frac{2}{2n-1}-1=\frac{2n}{2n-1}-1=\frac{1}{2n-1}  \\ \\ $
$\sum_{n=1}^{+\infty}\frac{1}{2n-1} $  diverges  ,then $\sum_{n=1}^{+\infty}a_n $ diverges 
but according to Wolfram  $\sum_{n=1}^{+\infty}a_n=\frac{1}{2}-\frac{\log 2}{2}$
where is the mistake in my solution please?
thanks

Your mistake was assuming this: If $x_n \sim y_n,$ then $x_n - 1 \sim y_n -1.$ A counterexample is $x_n =1+ 1/n, y_n = 1+1/n^2.$
