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(x)=n=1(n+1)nn!xn1, so you are looking for f(1);


  • For all xR, f(x)=xex using the known power series for exp, so that f(x)=(x+2)ex.





Therefore, f(1)=3e.


No comments:

Post a Comment

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find limh0sin(ha)h without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...