calculus - Sum of infinite series $1+frac22+frac3{2^2}+frac4{2^3}+cdots$

How do I find the sum of $\displaystyle 1+{2\over2} + {3\over2^2} + {4\over2^3} +\cdots$

I know the sum is $\sum_{n=0}^\infty (\frac{n+1}{2^n})$ and the common ratio is $(n+2)\over2(n+1)$ but i dont know how to continue from here

Answer

After establishing convergence, you could do the following:

$$S = 1 + \frac22 + \frac3{2^2}+\frac4{2^3}+\dots$$
$$\implies \frac12S = \frac12 + \frac2{2^2} + \frac3{2^3}+\frac4{2^4}+\dots$$
$$\implies S - \frac12S = 1+\frac12 + \frac1{2^2} + \frac1{2^3}+\dots$$
which is probably something you can recognise easily...