Summing a Geometric Series with Variable Exponents

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Homework Statement



1/2 + 2/4 + 3/8 + 4/16 + 5/32 + ...

Homework Equations





The Attempt at a Solution



The only headway I've made is that this is ∑ n/(2^n). how do I go about summing this?
 
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how about starting by adding up the first few terms and looking for a pattern with n terms
 
Define f(x)=x^n/2^n. If you find that then your sum just might be related to f'(x) at x=1.
 

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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