Tuesday, 11 July 2017

summation - Alternating series of reciprocals of factorials



Consider the sum:
(1)22!+(1)33!+(1)44!++(1)nn!.
Does there exist a nice closed form of such sum?



Answer



(UPDATED)
Concerning the finite sum :
Sn:=nk=2(1)kk!
we have simply : Sn=1n![n!e]=!nn!
with [x] the nearest integer (i.e. the round function) and !n the number of derangements for n elements
(added from Wood's answer (+1) in the related thread for additional properties).



To prove this directly you may use a method similar to the one proposed in this thread.



The idea is that the remaining terms of the Maclaurin expansion of e1n! (after the n first terms) are : (1)n+1(n+1)+(1)n+1(n+1)(n+2)+  and thus with absolute value bounded by 12 for n>1 that will disappear with the round function.



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