Geometric series
Problem 7.72 · easy
Does \( \displaystyle \sum_{n=1}^{\infty} - \left(-1\right)^{n - 1} \) converge? If so, find its sum.
- \[ -1 \]The first term, a.✓ Proved
- \[ -1 \]The common ratio, r.✓ Proved
- |r| = 1 ≥ 1, so the series diverges.
Answer \( \text{diverges} \)
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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 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 | the terms do not shrink (60th term checked) |
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/geometric_series, checked 2026-09-26 with SymPy 1.14.0.