Home -> Solved problems -> e irrational number

## Prove that $$e$$ is an irrational number

### Solution

To start, let’s expand the exponential function
$\mathrm{e}^{x}=\sum_{k=0}^{\infty} \frac{x^{k}}{k !}$
$\Rightarrow e=\sum_{k=0}^{\infty} \frac{1}{k !}=1+\frac{1}{1 !}+\frac{1}{2 !}+\frac{1}{3 !}+\cdots$
Assume that e is rational
$e=\frac{m}{n}=1+\frac{1}{1 !}+\frac{1}{2 !}+\frac{1}{3 !}+\cdots \quad \text { for two coprime integers } m \text { and } n$
$\frac{m}{n}=1+\frac{1}{1 !}+\frac{1}{2 !}+\frac{1}{3 !}+\cdots+\frac{1}{n !}+\frac{1}{(n+1) !}+\frac{1}{(n+2) !}+\cdots$
$n ! \frac{m}{n}=n !+\frac{n !}{1 !}+\frac{n !}{2 !}+\frac{n !}{3 !}+\cdots+\frac{n !}{n !}+\frac{n !}{(n+1) !}+\frac{n !}{(n+2) !}+\cdots$
\begin{aligned} &\mathrm{A}=n ! \frac{m}{n}=(n-1) ! m \in \mathbb{N} \\ &\mathrm{B}=n !+\frac{n !}{1 !}+\frac{n !}{2 !}+\frac{n !}{3 !}+\cdots+\frac{n !}{n !} \in \mathbb{N} \\ &\mathrm{C}=\frac{1}{n+1}+\frac{1}{(n+1)(n+2)}+\frac{1}{(n+1)(n+2)(n+3)}+\cdots \end{aligned}
$\begin{gathered} \text { For } n \geqslant 1 \\ \frac{1}{n+1} \leqslant \frac{1}{2} \\ \frac{1}{(m+1)(n+2)}<\frac{1}{2^{2}} \\ \frac{1}{(n+1)(n+2)(n+3)}<\frac{1}{2^{3}} \\ \vdots \end{gathered}$
$\Rightarrow\mathrm{C}<\frac{1}{2}+\frac{1}{2^{2}}+\frac{1}{2^{3}}+\cdots=\left(\frac{1}{2}\right)\left(\frac{1}{1-\frac{1}{2}}\right)=1$
$0<c<1 \Rightarrow C \notin \mathbb{N}$
$\text { Therefore }$
$\huge \mathrm{e} \notin \mathbb{Q}$
Home -> Solved problems -> e irrational number

### Related Topics

Find the volume of the square pyramid as a function of $$a$$ and $$H$$ by slicing method.
Prove that $\lim_{x \rightarrow 0}\frac{\sin x}{x}=1$
Prove that
Calculate the half derivative of $$x$$
Prove Wallis Product Using Integration
Calculate the radius R
Calculate the volume of Torus using cylindrical shells
Find the derivative of exponential $$x$$ from first principles
Calculate the sum of areas of the three squares
Find the equation of the curve formed by a cable suspended between two points at the same height
Solve the equation for real values of $$x$$
Solve the equation for $$x\epsilon\mathbb{R}$$
Determine the angle $$x$$
Calculate the following limit
Calculate the following limit
Calculate the integral
Challenging problem
Prove that
Prove that $$e$$ is an irrational number
Home -> Solved problems -> e irrational number