Home›Calculus 2›Summing a series by differentiating or integrating›Problem 7.619

Summing a series by differentiating or integrating

Problem 7.619 · hard

Find the sum of \( \displaystyle \sum_{n=1}^{\infty} 2^{- n} n \).

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
\[ \left. x \frac{d}{d x} 1 \frac{1}{1 - x} \right|_{\substack{ x=1 \cdot \frac{1}{2} }} = 2 \]
Put x = 1/2.✓ Proved
\[ \sum_{n=1}^{\infty} 2^{- n} n = 2 \]
Summed directly, the series agrees.✓ Proved

Answer \( 2 \)

✓ 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 identifies the generating function for the series and applies the standard technique of differentiating the geometric series. The substitution and final result are correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the generating function for the series and applies the standard technique of differentiating the geometric series. The substitution and final result are correct.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the generating function for the series and applies the standard technique of differentiating the geometric series. The substitution of x=1/2 is valid as it lies within the radius of convergence, and the final result is correct.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly derives the generating function for the series and evaluates it at the specified point. The steps are logically sound and the final answer is correct.
gpt-oss:20b: pass 2026-10-14

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 2026-10-17 with SymPy 1.14.0.