On value of a serie $\sum_{n\geq0}\frac{1}{\alpha a^n+\beta b^n}$ Let $\alpha,\beta,a,b$ are non zero real numbers such that $a,b>1$ and  $\alpha a^n+\beta b^n\neq0$ with $n\in\mathbb{N}$. Then, the serie
$$\sum\limits_{n=0}^\infty \frac{1}{\alpha a^n+\beta b^n}$$
is convergent. What is the exact value of this series?

If you are concerned by$$S=\sum_{n=0}^\infty\frac{1}{\alpha\, a^n+\beta\, b^n}$$ the result is bounded by
$$\frac{a}{(a-1) (\alpha +\beta )}\qquad \text{and} \qquad \frac{b}{(b-1) (\alpha +\beta )}$$ Otherwise, as said in comments, assuming $b <a$ for example, write
$$S=\frac 1 \alpha\sum_{n=0}^\infty\frac{1}{ a^n}\frac{1}{1+\frac \beta \alpha\,  \left(\frac{b}{a}\right)^n}=\frac 1 \alpha \sum_{n=0}^\infty\frac{1}{ a^n}\frac{1}{1+\gamma\,  x^n}$$
$$S=\frac 1 \alpha \sum_{n=0}^\infty\frac{1}{ a^n}\sum_{k=0}^\infty (-1) ^k \gamma ^k x^{kn}$$
