sequences and series - How to prove $sumlimits_{i=1}^{infty}frac{i}{2^i}$ converges?

What would be the simplest way to prove that $\sum\limits_{i=1}^{\infty}\dfrac{i}{2^i}$ converges?

Answer

Another possible way : consider $$\sum\limits_{i=1}^{\infty}{i}{x^i}=x\sum\limits_{i=1}^{\infty}{i}{x^{i-1}}=x \frac{d}{dx}\Big(\sum\limits_{i=1}^{\infty}{x^i}\Big)=x \frac{d}{dx}\Big(\frac{x}{1-x}\Big)=\frac{x}{(1-x)^2}$$ Now, replace $x$ by $\frac{1}{2}$ to get $2$.

The same procedure would apply to $$\sum\limits_{i=1}^{\infty}\dfrac{i}{a^i}=\frac{a}{(a-1)^2}$$