- #1
namu
- 33
- 0
Homework Statement
Find the limit
[itex]
lim_{n \to \infty} \sum_{j=1}^n \frac{b^j}{(j+1)!}
[/itex]
Homework Equations
Geometric series sum:
[itex]
S=\sum_{j=1}^n r^n
[/itex]
[itex]
S-rS=(1-r)S=1-r^{n+1}
[/itex]
[itex]
S=\frac{1-r^{n+1}}{1-r}
[/itex]
[itex]
S \to \frac{1}{1-r} \,\,\, as \,\,\, n \to \infty
[/itex]
if [itex] |r|<1 [/itex]
The Attempt at a Solution
[itex]
b\sum_{j=1}^n \frac{b^j}{(j+1)!}-\sum_{j=1}^n \frac{b^j}{(j+1)!}=-\frac{b}{2}+\frac{b^2}{3}+\frac{b^3}{8}+...
[/itex]
I tried to use something similar as when deriving the sum of a geometric series, however was unsucessful. I don't know how to integrate a factorial, so I can't use that approach either. Does anyone have any suggestions?