Calculate cauchy product of series

Question

These are the series I need to find the Cauchy product to:

$$\sum_{n=0}^\infty q^n$$

and

$$\sum_{n=0}^\infty nq^n$$

Is it just

$$\sum_{j=0}^\infty\sum_{k=0}^j q_k^nnq^{n_ {j-k}}$$

or what am I missing? To be honest, I'm fairly confused about the concept of the Cauchy Product.

Answer 1

The product of the two series is given by

$$ \sum_{n=0}^{\infty} c_n , $$

where

$$ c_n = \sum_{i=0}^{n} iq^i q^{n-i} . $$

Now, you need to simplify $c_n$ and subs back in the first sum.

Note:

$$ \sum_{i=0}^{n} i = \frac{n(n+1)}{2} .$$