What is the closed form of $\sum_{i=1}^n\frac{2i}{2^i}$

Question

I looked at $\frac{\sum 2i}{\sum2^i}$ (division), however both expressions are not equal. I am looking for an expression like $\sum_{i=1}^n\frac{2i}{2^i}=5n$ for example.

Answer 1

Hint: Try to compare your expression with $f\left(\frac{1}{2}\right)$ where $f(x)=\frac{d}{dx}\left(x+x^{2}+...+x^{n}\right)$

Answer 2

Obviously you want solution for the second expression, I misunderstood you question. So for that you can write this as a value of a serie or troncated series

$$ f(x)=\sum_{i=0}^{+\infty \ \text{OR} \ N}a_ix^i$$

That you know the explicit expression.

You know that after derivation and variable change

$$ f'(x)= \sum_{i=0}^{+\infty \ \text{OR} \ N}a_{i+1}(i+1)x^i $$

Remark that you can continue the process for higher order.

For your problem, take a particular value of $f'(x)$ (here is 1/2).