comparison test and limit comparison test for series

[iDoof]

i've got this infinite series:

the SUM of (n!/(n^n)) from n=1 to infinity. i think one of the comparison test is what i need...but i don't know what to compare it to...


someone please lead the way for me!!!


thanks

 

[tkhunny]

What is it that you are trying to show? A problem statement would help.

 

[soroban]

Hello, iDoof!

I assume you're trying to prove convergence or divergence.

the SUM of (n!/(n^n)) from n=1 to infinity.

The Ratio Test works for me . . . but it's messy.

. . . . . .a_n+1 . . . . . .(n+1)! . . . nⁿ
R . = . ------- . = . ------------- . ----
. . . . . . .a_n . . . . . (n+1)ⁿ⁺¹ . . .n!


. . . . . . . . . . . . . . . nⁿ . . . . . . . . . . .nⁿ
. . . = . (n + 1) ---------------- . = . ----------
. . . . . . . . . . . .(n+1)(n+1)ⁿ . . . . (n + 1)ⁿ


. . . . . . . . . .1 . . . . . . . . . . .1
. . . = . -------------- . = . ------------
. . . . . . .(n + 1)ⁿ . . . . . . (n + 1)ⁿ
. . . . . . .---------- . . . . . . (-------)
. . . . . . . . . nⁿ . . . . . . . . ( . n . )

. . . . . . . . . 1
. . . = . -------------
. . . . . . (1 + 1/n)ⁿ


Now we take the limit as n -> infinity.
If you recall the definition of e, you see that this limit approaches 1/e.

Since: . lim R .= .1/e .< .1, the series converges.
. . . . . .n->oo