[enter image description here][1]
$$\sum_{k=1}^{\infty} k[(1-p)^k +p^k(1 -p)] =(1-p)p \sum_{k=1}^{\infty} [k(1-p)^{k-1} +\sum_{k=1}^{\infty} kp^{k-1}]$$
$$=(1-p)p + [\frac{1}{p^2} +\frac{1}{(1-p)^2}] =\frac{1-2p+2p^2}{p(1-p)}$$
Can someone please let me know how the 2 summed series simplified to $\frac{1}{p^2}$ and $\frac{1}{p-1}^2$
I suspect it to be a geometric series and I seem to have $\frac{1}{p}-\frac{1}{1-p}$
