- #1
Contingency
- 41
- 0
Homework Statement
Does [tex]\sum _{ n=1 }^{ \infty }{ \frac { { \alpha }^{ n }{ n }! }{ { n }^{ n } } } [/tex] converge [tex]\forall |\alpha |<e[/tex]
and if so, how can I prove it?
Homework Equations
[tex]{ e }^{ x }=\sum _{ n=0 }^{ \infty }{ \frac { { x }^{ n } }{ n! } } [/tex]
The Attempt at a Solution
In the case of e I get the strange and probably wrong [tex]\sum _{ n=1 }^{ \infty }{ \frac { { (\sum _{ n=0 }^{ \infty }{ \frac { 1 }{ n! } } ) }^{ n }{ n }! }{ { n }^{ n } } } [/tex]
which seems to be the boundary between convergence and divergence for the series, but I have no idea how to actually make anything of this.