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Find the sum 1/2 + 2/2^2 + 3/2^3 + 4/2^4 + 5/2^5 + ...

 Dec 16, 2019
 #1
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0

∑[n / 2^n, n, 1, ∞] = 2

 Dec 16, 2019
 #2
avatar+23809 
+3

Find the sum = \(\dfrac{1}{2} + 2*\dfrac{1}{2^2} + 3*\dfrac{1}{2^3} + 4*\dfrac{1}{2^4} + 5*\dfrac{1}{2^5}+ \ldots\)

 

\(\begin{array}{rcrl} s &=& 1*\dfrac{1}{2} +& 2*\dfrac{1}{2^2} + 3*\dfrac{1}{2^3} + 4*\dfrac{1}{2^4} + 5*\dfrac{1}{2^5}+ \ldots \\ \dfrac{1}{2}s &=& & 1*\dfrac{1}{2^2} + 2*\dfrac{1}{2^3} + 3*\dfrac{1}{2^4} + 4*\dfrac{1}{2^5}+ \ldots \\ \hline s-\dfrac{1}{2}s &=& 1*\dfrac{1}{2} +& 1*\dfrac{1}{2^2}+1*\dfrac{1}{2^3} +1*\dfrac{1}{2^4} +1*\dfrac{1}{2^5}+ \ldots \\ \end{array} \)

\(\begin{array}{rcll} \dfrac{1}{2}s &=& \dfrac{1}{2}* \left( 1+ \dfrac{1}{2^1}+1*\dfrac{1}{2^2} +1*\dfrac{1}{2^3} +1*\dfrac{1}{2^4}+ \ldots \right) \\ s &=& 1+ \dfrac{1}{2^1}+1*\dfrac{1}{2^2} +1*\dfrac{1}{2^3} +1*\dfrac{1}{2^4}+ \ldots \\ s &=& 1+ \dfrac{1}{2^1}+\dfrac{1}{2^2} +\dfrac{1}{2^3} + \dfrac{1}{2^4}+ \ldots \\ \end{array}\)

 

Now use the formula for the sum of an infinite geometric series.

\(\begin{array}{|rcll|} \hline s &=& \dfrac{1}{1-\dfrac{1}{2}} \\ s &=& \dfrac{1}{ \dfrac{1}{2}} \\ \mathbf{s} &=& \mathbf{2} \\ \hline \end{array}\)

 

laugh

 Dec 16, 2019
 #3
avatar+106519 
+1

Very nice, heureka   !!!!!

 

 

cool cool cool

CPhill  Dec 16, 2019
 #4
avatar+23809 
+1

Thank you, CPhill !

 

laugh

heureka  Dec 16, 2019

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