Series summation of Geometric-Harmonic series

Question

I am trying to find the series summation for the following series :

$ \sum_{i=0}^{k} \frac{\beta^i}{i+1}$

and

$ \sum_{i=0}^{k} \frac{\beta^i}{(i+1)^2}$

$\beta \in (0,1)$

Any ideas on how to proceed? Or are there any existing results in this regard?

Answer 1

Let $$S_1(\beta)= \sum_{i=0}^{k} \frac{\beta^i}{i+1}.$$ Multiplying by $\beta$ and deriving term-wise,

$$(\beta S_1(\beta))'=\sum_{i=0}^{k} \beta^i=\frac{1-\beta^{k+1}}{1-\beta}.$$

Now you can integrate by means of the incomplete Beta function, or its hypergeometric representation,

$$\beta S_1(\beta)=-\ln(1-\beta)-B(\beta;k+2,0)=-\ln(1-\beta)-\frac{\beta^{k+1}}{k+1}F(k+1,1,k+2;\beta).$$

Nothing really appetizing.

Similarly,

$$(\beta S_2(\beta))'=S_1(\beta).$$