<s>[INST] 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
 [/INST] Once you've determined that the sum converges (you can do this by the ratio test or by integration as other users have pointed out), you can find the value quite nicely. Let 
$$S=1+\frac{2}{2}+\frac{3}{4}+\frac{4}{8}+\frac{5}{16}+\cdots$$
Then
$$2S=2+\frac{2}{1}+\frac{3}{2}+\frac{4}{4}+\frac{5}{8}+\frac{6}{16}+\cdots$$
From the second term on, this shares denominators with $S$ itself, so we can write
$$2S-S=2+1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots$$
But this is just a geometric series, so we get
$$S=\boxed{4}$$
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