Geometric series
Problem 7.2 · medium
Does \( \displaystyle \sum_{n=0}^{\infty} - \left(- \frac{3}{5}\right)^{n} \) converge? If so, find its sum.
- \[ -1 \]The first term, a.✓ Proved
- \[ - \frac{3}{5} \]The common ratio, r.✓ Proved
- |r| = 3/5 < 1, so the series converges to a/(1 − r).
- \[ - \frac{5}{8} \]a/(1 − r).✓ Proved
Answer \( - \frac{5}{8} \)
Lines: 3 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 |
| 4 | ✓ 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 | the 400th partial sum 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/geometric_series, checked 2026-09-26 with SymPy 1.14.0.