Given:
$\sum_{k = 1}^\infty\dfrac{6^k}{(3^k - 2^k)(3^{k+1} - 2^{k+1})}$
The Partial Fractions Decomposition is:
$\sum_{k = 1}^\infty(\dfrac{3^k}{3^k - 2^k} - \dfrac{3^{k+1}}{3^{k+1} - 2^{k+1}})$
But I cannot derive the decomposition myself, no matter what I try. Could someone therefore provide the steps to calculate the decomposition along with their reasoning?
write $$\begin{align*}6^k &= (6^k)(3-2) \\&= 3^{k+1}2^k - 3^k2^{k+1}\\ &= 3^{k+1}2^k - 3^k2^{k+1} + 3^{2k+1} - 3^{2k+1}\end{align*}$$
take (first and fourth) and (second and third) term and factorise
it will split into the req form