Determine the radius of convergence and test convergence
Determine the radius of convergence and test convergence at the endpoints for $\displaystyle\sum_{n=1}^{\infty}\frac{x^n}{n!}$.
Explanation of Method
Factorial-decaying coefficients give an infinite radius of convergence by the ratio test, since the ratio of successive coefficients tends to $0$ regardless of $x$. There are no finite endpoints to test.
- Apply the ratio test to find $R$.
- Note there are no finite endpoints since $R=\infty$.
- Identify the closed form as $e^x-1$.
Step 1 · Radius of convergence
$$ \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| = \lim_{n\to\infty}\frac{n!}{(n+1)!} = \lim_{n\to\infty}\frac1{n+1} = 0 \implies R=\infty. $$ The series converges absolutely for every $x\in\mathbb R$.
Step 2 · Endpoints
Since $R=\infty$, there are no finite endpoints to test — the series converges on all of $(-\infty,\infty)$.
Step 3 · Closed form
Since $e^x=\sum_{n=0}^{\infty}x^n/n!=1+\sum_{n=1}^{\infty}x^n/n!$: $$ \sum_{n=1}^{\infty}\frac{x^n}{n!} = e^x-1. $$
Summary Table
| Property | Value |
|---|---|
| Radius of convergence | $R=\infty$ |
| Interval of convergence | $(-\infty,\infty)$ |
| Closed form | $e^x-1$ |
Additional Notes
- This is the fastest-converging coefficient sequence among all the power series in this batch — factorial decay in the denominator beats any exponential or polynomial growth in the numerator, exactly the mechanism behind Problem 202's and Problem 233's uniform-convergence-on-bounded-sets results.
- Since $R=\infty$, there is no boundary behavior to analyze — a sharp contrast with every other problem in this batch, all of which have finite radius and non-trivial endpoint questions.
- This series is the base case from which Problem 233's much more delicate radius $1/e$ is derived: multiplying the $1/n!$ coefficient by the rapidly growing factor $(1+1/n)^{n^2}$ shrinks the radius from infinite down to a finite value.