There are many well-known methods for efficiently numerically computing $\pi$, such as Chudnovsky's Method or perhaps Gauss-Legendre's algorithm. I was wondering what the best method for computing $e$ is.
I have used the Taylor's series expansion or $e^x$ at $x=1$, which is in fact very convergent and does not cost too much to compute:
$e=\sum_{n=0}^{\infty}\frac{1}{n!} = 1+1+\frac{1}{2}+\frac{1}{6}+\cdots$
Here are a table of some basic values:
n Estimation Error (e - sum) 1 1.0 1.718281828459045 5 2.708333333333333 0.009948495125712054 10 2.7182815255731922 3.0288585284310443E-7 100 2.7182818284590455 -4.440892098500626E-16
I am curious to know if there are even more efficient methods to compute $e$. Keep in mind that computational cost and convergence speed are the priorities.