Using D'Alembert's ratio test, determine the convergence
Using D'Alembert's ratio test, determine the convergence of: $$ \sum_{n=1}^{\infty} \frac{1}{n!}. $$
Explanation of Method
The ratio test compares consecutive terms directly: $\frac{a_{n+1}}{a_n}=\frac1{n+1}\to0$, well below the convergence threshold of $1$, since factorials in the denominator make each successive term dramatically smaller than the last. This is the standard route to recognizing $\sum1/n!$ as the (shifted) Taylor series of $e^x$ at $x=1$.
- State D'Alembert's ratio test.
- Compute the ratio $a_{n+1}/a_n$ and its limit.
- Apply the test and identify the sum in terms of $e$.
Step 1 · State the ratio test
For $a_n>0$, compute $L=\lim_{n\to\infty}\frac{a_{n+1}}{a_n}$: if $L<1$ the series converges, if $L>1$ it diverges, and $L=1$ is inconclusive.
Step 2 · Compute the ratio
With $a_n=\dfrac1{n!}$:
$$ \frac{a_{n+1}}{a_n} = \frac{n!}{(n+1)!} = \frac1{n+1} \longrightarrow 0. $$
Step 3 · Apply the test
Since $L=0<1$, by D'Alembert's ratio test $\sum_{n=1}^{\infty}\dfrac1{n!}$ converges. Since $e=\sum_{n=0}^{\infty}\frac1{n!}$, subtracting the $n=0$ term ($=1$):
$$ \sum_{n=1}^{\infty}\frac1{n!} = e-1 \approx 1.71828. $$
Summary Table
| Quantity | Value |
|---|---|
| Ratio $a_{n+1}/a_n$ | $\dfrac1{n+1}$ |
| $L=\lim a_{n+1}/a_n$ | $0$ |
| Sum | $e-1\approx1.71828$ |
Additional Notes
- Because the ratio $\frac1{n+1}\to0$ rather than merely staying below $1$, convergence here is extremely fast — the terms shrink super-geometrically, so partial sums reach $e-1$ to many decimal places within just a handful of terms (e.g. $S_{10}\approx1.7182818$, correct to 7 digits).
- This series is the tail of the defining Taylor expansion $e^x=\sum_{n=0}^{\infty}x^n/n!$ evaluated at $x=1$, so its rapid convergence is exactly why $n!$ in a denominator is the standard device for constructing entire functions with infinite radius of convergence.
- This result underlies Problems 60 and 62, both of which show that $\sum a^n/n!$ and $\sum n^p/n!$ converge for every $a>0$ and every real $p$ — the same mechanism (ratio $\to0$) is at work in each case.
Result
The series converges, with sum: