2007-12-11 15:22:07 · 4 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
the series converges because each answer gets progressively smaller because n! gets bigger. This series is part of a series that converges to e namely: 1+1/1!+1/2!+1/3!+1/4!... . Hope this helps!
2007-12-11 15:35:23 · answer #1 · answered by Nati F 3 · 0⤊ 7⤋
1 N Converge Or Diverge
2016-11-07 05:46:59 · answer #2 · answered by ritzer 4 · 0⤊ 0⤋
It converges. One way to see it is via the ratio test: Let a_n=1/n!
a_(n+1)/a_n=n!/(n+1)!. Since (n+1)!=(n+1)*n!, we get that a_(n+1)/a_n=1/(n+1) which goes to 0 as n goes to infinity. Since 0<1, the series converges.
A second proof can be obtained via the comparison test. For n>1, n!>=n(n-1), so a_n<=1/[n(n-1)]. Now the series Σ1/n(n-1) converges, because the series Σ1/n^2 converges (p- series, p>1) and you can see this via the limit comparison test.
2007-12-11 15:32:46 · answer #3 · answered by partalopoulo 2 · 12⤊ 0⤋
The series 1/n!, for n = 0 to infinity, converges to e (2.718281828...).
2007-12-11 15:27:53 · answer #4 · answered by lithiumdeuteride 7 · 1⤊ 2⤋