Wednesday, 31 August 2016

summation - Evaluate the sum of this series





Please help me find the sum of this series:



1+2312+2356122+235689123+



All I could figure out was to find the nth term as:



an=2(2+3)(2+3(n1))3693(n1)12n1



What To do ahead of it. I don't know. Please help.


Answer




Let S denote the sum. We write each term (with indices starting at 0) as



(nk=13k13k)12n=n1k=0(23k)n!(12)n=(2/3n)(12)n.



Then we easily recognize S as a bionmial series and hence



S=n=0(2/3n)(12)n=(112)2/3=22/3.


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