# Thread: limit of a series

1. ## limit of a series

$y_n=\frac{t^{n+1}}{(n+1)!}+2\sum_{k=0}^{n}\frac{t^ {k}}{k!}-t-1$
what is the limit of this series when n goes to infinity
?

2. Originally Posted by transgalactic
$y_n=\frac{t^{n+1}}{(n+1)!}+2\sum_{k=0}^{n}\frac{t^ {k}}{k!}-t-1$
what is the limit of this series when n goes to infinity
?
For any fixed $t \in \mathbb{R}$

$\displaystyle \lim_{n \to \infty}\frac{t^{n+1}}{(n+1)!}=0$

$\displaystyle \lim_{n \to \infty}\sum_{k=0}^{n}\frac{t^k}{k!}=e^t$

This is the Taylor series of $e^t$ centered a 0.

3. Originally Posted by TheEmptySet
For any fixed $t \in \mathbb{R}$

$\displaystyle \lim_{n \to \infty}\frac{t^{n+1}}{(n+1)!}=0$

why its zero i dont know whats t?
the denominetor could be zero too
$\displaystyle \lim_{n \to \infty}\sum_{k=0}^{n}\frac{t^k}{k!}=e^t$

This is the Taylor series of $e^t$ centered a 0.
how to prove that its exponent series
i cant write "its the same"

4. $\displaystyle f(t)=e^t \implies \frac{d^n}{dt^n}f(t)=e^t$ for all $n$.
$\displaystyle \frac{d^n}{dt^n}f(t)\bigg|_{t=0}=1$ for all $n$
$\displaystyle \sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}t^n=\sum_{ n=0}^{\infty}\frac{t^n}{n!}$
Since $-t-1$ does not depend on n what is the limit
$\displaystyle \lim_{n \to \infty}(-t-1)=(-t-1)\lim_{n \to \infty}1=?$