+26367 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.
\(\begin{array}{|rcll|} \hline s &=& \dfrac{1}{1-\dfrac{1}{2}} \\ s &=& \dfrac{1}{ \dfrac{1}{2}} \\ \mathbf{s} &=& \mathbf{2} \\ \hline \end{array}\)
