Sunday, 1 May 2016

summation - Find the sum of the infinite series sumn(n+1)/n!



How do find the sum of the series till infinity?



21!+2+42!+2+4+63!+2+4+6+84!+



I know that it gets reduced to n=1n(n+1)n!
But I don't know how to proceed further.



Answer



Define f by f(x)=n=0xn+1n! for xR. (It is easy to check that the radius of convergence of this function is infinite.)



In particular:




  • For all xR, f, so you are looking for f''(1);


  • For all x\in\mathbb{R}, f(x) = x e^x using the known power series for \exp, so that f''(x) = (x+2)e^x.





Therefore, f''(1) = 3e.


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