Radius of convergence
Problem 7.53 · easy
Find the radius of convergence of \( \displaystyle \sum_{n=1}^{\infty} \frac{x^{n}}{n!} \).
- Use the ratio test on the absolute values of the terms.
- \[ \lim_{n \to \infty}\left(\frac{x^{- n} x^{n + 1} n!}{x \left(n + 1\right)!}\right) = 0 \]|a_(n+1)/a_n| → 0·|x − 0|.✓ Proved
- The series converges when 0|x − 0| < 1, which is every x: R = ∞.
Answer \( R = \infty \)
Lines: 1 proved, 2 not checked. The answer was also checked a second way, without looking at the solution. The explanation has not been reviewed yet.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | Not checked | — | a sentence; read, not computed |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | terms evaluated just inside and just outside the radius shrink and grow |
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/radius_of_convergence, checked 2026-09-26 with SymPy 1.14.0.