Sum of the reciprocals of the factorials

Example

The series $$\sum_{n=0}^{\infty} \frac{1}{n!} = 1 + 1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} + \cdots$$ converges, and its sum is the number e $\approx 2.71828$.

Remarks

The series converges by the ratio test, since the ratio of consecutive terms is $1/(n+1) \to 0$. It is the value at $x = 1$ of the power series $\sum x^n / n!$ of the exponential function. The series converges fast. The tail after the term $1/N!$ is less than $1/(N! \, N)$.

Explanations

L1 human

The terms shrink very fast, because $n!$ grows fast. Only a few terms give a good value of $e$: the first five terms already give $2.708$.