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