Problem D
Euler's Number
Euler’s number (you may know it better as just $e$) has a special place in mathematics. You may have encountered $e$ in calculus or economics (for computing compound interest), or perhaps as the base of the natural logarithm, $\ln {x}$, on your calculator.
While $e$ can be calculated as a limit, there is a good approximation that can be made using discrete mathematics. The formula for $e$ is:
\begin{align*} e & = \displaystyle \sum _{i=0}^ n\dfrac {1}{i!}\\ & = \dfrac {1}{0!} +\dfrac {1}{1!} +\dfrac {1}{2!}+\dfrac {1}{3!}+\dfrac {1}{4!} + \cdots \\ \end{align*}Note that $0! = 1$. Now as $n$ approaches $\infty $, the series converges to $e$. When $n$ is any positive constant, the formula serves as an approximation of the actual value of $e$. (For example, at $n=10$ the approximation is already accurate to $7$ decimals.)
You will be given a single input, a value of $n$, and your job is to compute the approximation of $e$ for that value of $n$.
Input
A single integer $n$, ranging from $0$ to $10\, 000$.
Output
A single real number – the approximation of $e$ computed by the formula with the given $n$. All output must be accurate to an absolute or relative error of at most $10^{-12}$.
| Sample Input 1 | Sample Output 1 |
|---|---|
3 |
2.6666666666666665 |
| Sample Input 2 | Sample Output 2 |
|---|---|
15 |
2.718281828458995 |
