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 x∈R. (It is easy to check that the radius of convergence of this function is infinite.)
In particular:
For all x∈R, f″(x)=∑∞n=1(n+1)nn!xn−1, so you are looking for f″(1);
For all x∈R, f(x)=xex using the known power series for exp, so that f″(x)=(x+2)ex.
Therefore, f″(1)=3e.
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