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Geometric Series Convergence

A public series question that checks the common-ratio convergence condition and applies the closed-form sum of an infinite geometric series.

Question

Evaluate ∑n=1∞4(25)n−1\displaystyle\sum_{n=1}^{\infty} 4\left(\tfrac{2}{5}\right)^{n-1}.

  1. a.83\dfrac{8}{3}
  2. b.207\dfrac{20}{7}
  3. c.203\dfrac{20}{3}
  4. d.The series diverges

Correct answer

C. 203\dfrac{20}{3}

Full reasoning

  1. 1Identify the geometric structure: the first term (n=1n = 1) is a=4a = 4 and each term is multiplied by r=25r = \tfrac{2}{5}.
  2. 2Check convergence: ∣r∣=25<1|r| = \tfrac{2}{5} < 1, so the infinite series converges.
  3. 3Apply the closed form: S=a1−r=41−2/5=43/5=203S = \dfrac{a}{1 - r} = \dfrac{4}{1 - 2/5} = \dfrac{4}{3/5} = \dfrac{20}{3}.

Why each choice is right or wrong

Choice A

This uses ar=85ar = \tfrac{8}{5} as the first term, effectively starting the sum from the second term of the series.

Choice B

This computes a1+r\dfrac{a}{1 + r}, flipping the sign in the denominator of the geometric sum formula.

Choice C

Correct. The first term is a=4a = 4 and the ratio is r=25r = \tfrac{2}{5} with ∣r∣<1|r| < 1, so the sum is 41−2/5=43/5=203\dfrac{4}{1 - 2/5} = \dfrac{4}{3/5} = \dfrac{20}{3}.

Choice D

This assumes adding infinitely many positive terms must diverge, but the ratio 25\tfrac{2}{5} satisfies ∣r∣<1|r| < 1, so the partial sums converge.

Related formula

Sum of an infinite geometric series

S=a1−r,∣r∣<1S = \frac{a}{1 - r}, \qquad |r| < 1

Assumptions: The common ratio satisfies |r| < 1; otherwise the series diverges.

Related topic and practice

Series Convergence TestsCalculus Domain GuideA limit at a missing pointCreate a free account to practice

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Sources

Authored for this public explainer registry. It has no database question ID and is not copied from a protected or official exam bank.

  1. Calculus Volume 2, Section 5.2: Infinite Series — OpenStax. Accessed 2026-08-03. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
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Sources last checked 2026-08-07

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