Telescoping series
Problem 7.324 · easy
Find the sum of \( \displaystyle \sum_{n=1}^{\infty} - \frac{1}{\sqrt{n + 1}} + \frac{1}{\sqrt{n}} \), or show that it diverges.
- \[ - \frac{1}{\sqrt{n + 1}} + 1 \frac{1}{\sqrt{n}} = - \frac{1}{\sqrt{n + 1}} + \frac{1}{\sqrt{n}} \]Partial fractions: the term is a difference.✓ Proved
- In the N-th partial sum every negative piece cancels a positive piece 1 term later; only the first positive and last negative pieces survive.Reviewed
- So S_N = 1 - 1/sqrt(N + 1).Reviewed
- \[ \lim_{k \to \infty}\left(1 - \frac{1}{\sqrt{k + 1}}\right) = 1 \]Let N → ∞.✓ Proved
Answer \( 1 \)
✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 3 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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 4000th partial sum, added term by term, equals 1 − 1/√4001, whose limit is 1 |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the series as telescoping and computes the limit of the partial sums accurately.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the series as telescoping and computes the limit of the partial sums accurately.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the series as telescoping, derives the correct partial sum formula, and computes the limit accurately.gpt-oss:20b: pass 2026-10-05
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/telescoping_series, checked 2026-10-05 with SymPy 1.14.0.