Saturday, 12 July 2014

convergence divergence - Value of this convergent series: frac13!+frac25!+frac37!+frac49!+cdots



What is the value of- 13!+25!+37!+49!+ I wrote it as general term n(2n+1)!. As the series converges it should be telescopic (my thought). But i dont know how to proceed. I also know 1n!=e Any help /hints appreciated. Thanks!


Answer



One may write
n=0n(2n+1)!=12n=0(2n+1)1(2n+1)!=12n=01(2n)!12n=01(2n+1)!=12(e+e12)12(ee12)=12e1


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