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how to add to infinity 1+2/2+3/2^2+4/2^3......Answer given is 4

how to add to infinity 1+2/2+3/2^2+4/2^3......Answer  given is 4

Grade:12th pass

3 Answers

Sourabh Singh IIT Patna
askIITians Faculty 2104 Points
7 years ago
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Mohammed Saleem Mohiuddin
23 Points
7 years ago
We can as well use the method of infinite sum of an arithmeticogemetric series.Here A.P. is 1,2,3,4.... and G.P.is 1,1/2,1/2^2,1/2^3........
a=1,d= 1, r=1/2,  Expression to  find infinite sum= a/(1-r)  +dr/(1-r)^2
mycroft holmes
272 Points
7 years ago
We can see that the general term of the expression is \frac{n}{2^{n-1}} = \frac{n+1}{2^{n-2} } - \frac{n+2}{2^{n-1} }
 
i.e. it can be written in the form f(n) – f(n+1) where f(n) = \frac{n+1}{2^{n-2} }
 
Hence we have \sum_{n=1}^{\infty} \frac{n}{2^{n-1}} =\sum_{n=1}^{\infty} \left(f(n)-f(n+1) \right ) = 4 - \lim_{n \rightarrow \infty}\frac{n+2}{2^{n-1}}
 
=\boxed{4}

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