Series convergence tests
Problem 7.77 · easy
Does \( \displaystyle \sum_{n=1}^{\infty} \frac{4^{n}}{n!} \) converge or diverge?
- Factorials suggest the ratio test.
- \[ \frac{4^{- n} 4^{n + 1} n!}{\left(n + 1\right)!} = \frac{4}{n + 1} \]a_(n+1)/a_n.✓ Proved
- \[ \lim_{n \to \infty}\left(\frac{4}{n + 1}\right) = 0 \]The limit is 0 < 1, so the series converges.✓ Proved
Answer \( \text{converges} \)
Lines: 2 proved, 1 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | SymPy's own convergence decision (Sum.is_convergent) agrees |
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/series_convergence, checked 2026-09-26 with SymPy 1.14.0.