How to simplify this convergent series?

Question

I was asked a question .After I got the answer,I was asked to simplify it. This is a part of the answer which was needed to be simplified: $\sum_{n=0}^{\infty}(n\,p^n)(1-p)^n+3(p^n)(1-p)^n$ I have considered about Bernoulli's principle and some of principles in my workbooks,and still have no idea about how to simplify it. Please someone come to help me or at least give me some hints! Thanks for coming to see my question!

Answer 1

Set $ x = p(1-p)$, then the sum is $$x\sum\limits_{n=1}^{\infty}n x^{n-1} + 3\sum\limits_{n=0}^{\infty}x^n$$

$$\sum\limits_{n=1}^{\infty}n x^{n-1} = \left(\sum\limits_{n=0}^{\infty}x^n\right)^{'}$$

Answer 2

If $p\in R$ then $ \sum_{0}^\infty (n+3)p^n(1-p)^n $ by ratio test we get $\lim_{n\to \infty }|\frac{(n+4)p(1-p)}{n}|=|p(1-p)|<1 \iff \frac{1-\sqrt{5}}{2}<p<\frac{1+\sqrt{5}}{2}$

And so, the series converges for only a certain set of values of p.