Infinite sum convergence $ \sum_{i\geq 1}\frac{1}{x^i-y^i}$

Question

For certain values of x and y, the sum $$\sum_{i=1}^{\infty}{\frac{1}{x^i-y^i}}$$ converges...is there a way to get the exact value, given x and y?

Answer 1

This is not precisely what you asked. (It goes from $-\infty$ to $\infty$ and must have $u \ne 1$.) But there is this very interesting series in terms of Jacobi theta functions... $$ \sum _{n=-\infty }^{\infty } \frac{1}{ {\alpha}^{n}-u{\beta}^{n} } =\frac{-i}{2}\,\frac{{ \theta_1} \left( \frac{1}{2}\,i\ln \left( {\frac { \alpha}{u}} \right) ,\sqrt {{\frac {\beta}{\alpha}}} \right) { \theta_2} \left( 0,\sqrt {{\frac {\beta}{\alpha}}} \right) { \theta_3} \left( 0,\sqrt {{\frac {\beta}{\alpha}}} \right) { \theta_4} \left( 0,\sqrt {{\frac {\beta}{\alpha}}} \right) }{ { \theta_1} \left( \frac{1}{2}\,i\ln \left( \alpha \right) , \sqrt {{\frac {\beta}{\alpha}}} \right) { \theta_1} \left( \frac{1}{2}\,i\ln \left( u \right) ,\sqrt {{\frac {\beta }{\alpha}}} \right) }$$