I need to find the sum of the following series as an explicit formula: $ \sum_{n=1}^\infty n(n+1)(\frac{1}{2})^n $. I can't express it as a geometric series $ \sum_{n=1}^\infty ar^{n-1} = \frac{a}{1-r} $ since the term n(n+1) is not constant.
I also considered that this series might be a Taylor expansion of a function $$ \sum_{n=1}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n $$ but I don't see how $ \frac{f^{(n)}}{n!} $ Could give me a constant term n(n+1). Does anyone have an idea of how I could proceed? Thanks!