Home›Calculus 2›Summing a series by differentiating or integrating›Problem 7.684

Summing a series by differentiating or integrating

Problem 7.684 · hard

Find the sum of \( \displaystyle \sum_{n=1}^{\infty} 4^{- n} n \).

Differentiate Σ xⁿ = 1/(1 − x) term by term and multiply by x: Σ n xⁿ = x/(1 − x)² for |x| < 1.
Reviewed
\[ \frac{d}{d x} \frac{1}{1 - x} = \frac{1}{\left(1 - x\right)^{2}} \]
d/dx of 1/(1 − x).✓ Proved
\[ \left. x \frac{d}{d x} 1 \frac{1}{1 - x} \right|_{\substack{ x=1 \cdot \frac{1}{4} }} = \frac{4}{9} \]
Put x = 1/4.✓ Proved
\[ \sum_{n=1}^{\infty} 4^{- n} n = \frac{4}{9} \]
Summed directly, the series agrees.✓ Proved

Answer \( \frac{4}{9} \)

✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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
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 first 399 terms added in floating point

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly derives the closed form for the arithmetico-geometric series and verifies the result. The steps are logically sound and the final answer is correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly derives the closed form for the arithmetico-geometric series and verifies the result. The steps are logically sound and the final answer is correct.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the series as a derivative of a geometric series and applies the standard formula Σnxⁿ = x/(1-x)² with x=1/4. The steps are logically sound and the result is correct.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly derives the generating function for the series and evaluates it at the appropriate point within the radius of convergence.
gpt-oss:20b: pass 2026-10-15

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_sum_by_calculus, checked 2026-10-18 with SymPy 1.14.0.