calculate the serie $\sum \frac{6^n}{(3^{n+1}-2^{n+1})(3^n - 2^n)}$ The following serie is obviously convergent, but I can not figure out how to calculate its sum :
$$\sum \frac{6^n}{(3^{n+1}-2^{n+1})(3^n - 2^n)}$$

You can use one of the following identities to telescope the series: $$\dfrac{3^{n+1}}{3^{n+1}-2^{n+1}}-\dfrac{3^n}{3^n-2^n}=\dfrac{2^n}{3^n-2^n}-\dfrac{2^{n+1}}{3^{n+1}-2^{n+1}}=\dfrac{6^n}{(3^{n+1}-2^{n+1})(3^n-2^n)}$$
