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

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In summary, the problem is asking to find the sum of the series n/2^n, from n=1 to infinity. Using the equation for the summation of x^n, which is 1/(1-x), we can differentiate it and put x=1/2 to get the solution. However, we may need to consider uniform convergence in order to justify differentiating under the sum. Alternatively, the problem can also be solved by rearranging the series without using the differentiation method.
  • #1
Spark15243
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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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  • #2
Have you covered uniform convergence of series?
 
  • #3
I...don't know. Ik that the terms themselves converge to zero, but I don't know how to calc the sum...
 
  • #4
Differentiate Summation(x^n)=1/(1-x). Put x=1/2.
 
  • #5
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.
 
  • #6
What do I do about the extra n in front then?
 

1. What is the formula for finding the sum of a series: n/2^n?

The formula for finding the sum of a series, n/2^n, is S = (n/2) / (1 - 1/2) = n.

2. How do you apply this formula to find the sum of a specific series?

To apply this formula to find the sum of a specific series, you first need to determine the value of n, which is the number of terms in the series. Then, plug in the value of n into the formula S = (n/2) / (1 - 1/2) = n to find the sum.

3. Can this formula be used for any series with n/2^n as the pattern?

Yes, this formula can be used for any series with n/2^n as the pattern. As long as the series follows this pattern, the formula will work to find the sum.

4. What does the value of n represent in this formula?

The value of n represents the number of terms in the series. This can also be thought of as the position of the last term in the series.

5. How can this formula be applied to real-world situations?

This formula can be applied to real-world situations in which there is a repeating pattern that follows the n/2^n pattern. For example, it could be used to calculate the total distance traveled by a car that travels half of its previous distance each time.

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