<s>[INST] Find the limit of $\lim_{n\rightarrow\infty}(\frac{1}{2}+\frac{3}{2^2}+...+\frac{2n-1}{2^n})$ Can anybody help me with this limit? 
$$\lim_{n\rightarrow\infty}\left(\frac{1}{2}+\frac{3}{2^2}+\cdots+\frac{2n-1}{2^n}\right)$$
Abel's transform wouldn't work on this one I guess, because another serie diverges
As this contains a geometric series with ratio $1/2$, I had multiplied whole series by $1/2$ and then subtracted from original, but got original series back. Any ideas how can the limit be calculated?
 [/INST] $S \ \ = \ \ \ \ \ \ \ \dfrac{1}{2}+\dfrac{3}{2^2}+\dfrac{5}{2^3}+\dfrac{7}{2^4}+\cdots$. 
$2S = 1+\dfrac{3}{2}+\dfrac{5}{2^2}+\dfrac{7}{2^3} + \dfrac{9}{2^4} + \cdots$
Now, subtract the first equation from the second, but match up terms with the same denominator: 
$S = 1+ \dfrac{2}{2} + \dfrac{2}{2^2} + \dfrac{2}{2^3} + \cdots$
This becomes $1$ plus a geometric series, which is easy to evaluate. 
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