How to prove that for $\displaystyle a>0$ and $\displaystyle s_j = \sum_{i=0}^j \frac{a^i}{i!}$ the series

$\displaystyle f_j = \frac{s_j}{s_{j-1}}$

is decreasing?

Thanks in advance!

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- Jul 20th 2012, 11:23 AMsanderDecreasing series
How to prove that for $\displaystyle a>0$ and $\displaystyle s_j = \sum_{i=0}^j \frac{a^i}{i!}$ the series

$\displaystyle f_j = \frac{s_j}{s_{j-1}}$

is decreasing?

Thanks in advance! - Jul 20th 2012, 10:55 PMProve ItRe: Decreasing series
- Jul 21st 2012, 07:37 AMHallsofIvyRe: Decreasing series
The series $\displaystyle \sum_{i=0}^\infty\frac{a^i}{i!}$ converges to $\displaystyle e^a$.