Finding the Sum of a Series: n/2^n

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Homework Help Overview

The problem involves finding the sum of the series defined by the expression n/2^n, with the summation extending from n = 1 to infinity. The subject area pertains to series convergence and manipulation.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the possibility of differentiating a known series sum to address the problem. There are questions about uniform convergence and how it relates to the manipulation of the series. Some express uncertainty about how to handle the extra n in the expression.

Discussion Status

The discussion is ongoing, with participants exploring different methods to approach the problem. Some guidance has been offered regarding differentiation of the series, but there is no explicit consensus on the best approach or resolution of the problem.

Contextual Notes

Participants are considering the implications of uniform convergence and the need for justification when manipulating the series. There is also a mention of previous experiences with similar problems, indicating varying levels of familiarity with the topic.

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Homework Statement



Find the sum of the series.

Summation (n/2^n)
from n = 1 to inf

Homework Equations



Summation (x^n) = 1/1-x

The Attempt at a Solution



n/2^n
n*(1/2)^n

Not sure what this equals after this... I was thinking (1/1-0.5) but that is a guess.
There are other ones like this, i have no idea how to do them, so any help on how to start and/or finish them will be appreciated.
 
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Have you covered uniform convergence of series?
 
I...don't know. Ik that the terms themselves converge to zero, but I don't know how to calc the sum...
 
Differentiate Summation(x^n)=1/(1-x). Put x=1/2.
 
This is what I had in mind, but don't we need uniform convergence to justify differentiating under the sum?

I remember doing this problem w/o this trick a year ago. It involved rearanging the series if my memory serves me right.
 
What do I do about the extra n in front then?
 

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