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Geometric Series Convergence: Public Worked Question

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 TestsSign in for current practice

This public explainer is separate from the protected practice bank. Current practice coverage varies.

Source and public-release record

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.

Review and maintenance

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Keiko Study editorial owner
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Last source check
2026-08-07
Next scheduled review
2026-11-07

Recheck the cited OpenStax reference before each major preparation cycle.

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