Welcome to mathimatikoi.org forum; Enjoy your visit here.

## Multiplicity of root

Real & Complex Analysis, Calculus & Multivariate Calculus, Functional Analysis,
Tolaso J Kos
Articles: 2
Posts: 860
Joined: Sat Nov 07, 2015 6:12 pm
Location: Larisa
Contact:

### Multiplicity of root

Given the function $f(x)=e^x-x-1$ prove that $0$ is a zero of $f$ of multiplicity $2$.
Imagination is much more important than knowledge.
Riemann
Articles: 0
Posts: 170
Joined: Sat Nov 14, 2015 6:32 am
Location: Melbourne, Australia

### Re: Multiplicity of root

Tolaso J Kos wrote:Given the function $f(x)=e^x-x-1$ prove that $0$ is a zero of $f$ of multiplicity $2$.
It suffices to prove that the limit $\displaystyle \lim \limits_{x \rightarrow 0} \frac{f(x)}{x^2}$ is finite. However,

\begin{align*}
\lim_{x\rightarrow 0} \frac{f(x)}{x^2} &= \lim_{x\rightarrow 0} \frac{e^x-x-1}{x^2} \\
&=\lim_{x\rightarrow 0} \frac{e^x-1}{2x} \\
&= \frac{1}{2}\lim_{x\rightarrow 0} e^x \\
&= \frac{1}{2}
\end{align*}

Done!
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$