# Numerical Analysis

\(e = \lim_{n\to\infty}(1+1/n)^n\)
The value of $e$ is approximated using the infinitely differentiable exponential function $\exp(x)$ and the **Taylorâ€™s Theorem**. This series can be used to obtain the value to a required degree of precision. Note that $c\in (x,a)$.
\(f(x) = f(a)+f'(a)(x-a) +\cdots+\frac{f^k(a)}{k!}(x-a)^k + \frac{f^{k+1}(c)}{(k+1)!}(x-a)^{k+1}\)
The final term when approximating the value of $e$ would be:
\(\frac{e^c}{(n+1)!}\)
We know that $e^c<3$ when $c\in (0,1)$. We can thus obtain the value of $e$ to a required degree of error by choosing an appropriate value of $n$. We can similarly compute the value of $e^a$ to any required degree of accuracy.