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:=n∑k=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 e−1n! (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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