<s>[INST] How can i find the sum? How can i calculate this serias?
$$\sum_{i=1}^n \frac{2^m}{2^i}\cdot i$$
I tried to do:
$$\ 2^m\cdot\left[\frac{1}{2}+\frac{2}{2^2}+...+\frac{n}{2^n}\right]$$
And i don't know how to continue..
Thank you.
 [/INST] $$s(n)=\frac{1}{2}+\frac{2}{2^2}+...+\frac{n}{2^n}\quad (1)$$
multiply by $1/2$ and get
$$\frac{s(n)}{2}=\frac{1}{2^2}+\frac{2}{2^3}+...+\frac{n-1}{2^{n}}+\frac{n}{2^{n+1}}\quad (2)$$
Now make $(1)-(2)$ and get
$$\frac{s(n)}{2}=\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{n}}-\frac{n}{2^{n+1}}\\
s(n)=\left[\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{n-1}}\right]-\frac{n}{2^{n}}$$
Can you finish?
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