Sum of the fractional series

Question

I've got a series and I wanted to calculate the sum. Thanks for help. The series is

$${\sum _{n=1}^{\infty } \frac{a_n}{b_n}}$$

Lets suppose that

$${\sum _{n=1}^{\infty }{a_n}}=C$$

$${\sum _{n=1}^{\infty }{b_n}}=D$$

Can I calculate with known constants $C$ and $D$ this series?

Answer 1

This is impossible in general form. Consider $a_n=\frac{1}{4^n}$ and $b_n=\frac{1}{2^n}-\frac{1}{3^n}$. Therefore $C=\frac{1}{3}$ and $D=\frac{1}{2}$ while there is no known closed form for $\sum_{n=1}^{\infty}\frac{a_n}{b_n}=\sum_{n=1}^{\infty}\frac{6^n}{12^n-8^n}$