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Worked problem · Calculus II · Units 3B–4B

Root Test Example: Worked Example and Solution

A complete course synthesis example with method choice, derivation, verification, and a common wrong approach.

1 problem8 min estimated timeCumulative progression

Problem

  1. Test ∑n=1∞(2n/(3n+1))n\sum_{n=1}^{\infty}(2n/(3n+1))^n.

Complete worked solutions

Every problem has a source-matched answer and independently reviewed derivation.

01

Problem 1: Test ∑n=1∞(2n/(3n+1))n\sum_{n=1}^{\infty}(2n/(3n+1))^n.

Answer: converges\text{converges}

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

  1. The root-test limit is lim⁡2n/(3n+1)=2/3<1\lim 2n/(3n+1)=2/3<1, so the series converges absolutely.
  2. The verified result is converges\text{converges}.

What is included

See a concise answer and full derivation for test sum from n equals 1 to infinity (2 n / (3 n plus 1)) to the power (n).

Skills assessed

  • Course synthesis

Prerequisites

  • Calculus II Units 3B through 4B

Common errors

  • Selecting a familiar formula without checking its hypotheses.

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Supporting steps