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

Limit Comparison 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∞(3n+1)/(n2+2)\sum_{n=1}^{\infty}(3n+1)/(n^2+2).

Complete worked solutions

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

01

Problem 1: Test ∑n=1∞(3n+1)/(n2+2)\sum_{n=1}^{\infty}(3n+1)/(n^2+2).

Answer: diverges\text{diverges}

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

  1. With bn=1/nb_n=1/n, an/bn=n(3n+1)/(n2+2)→3a_n/b_n=n(3n+1)/(n^2+2)\to3; limit comparison with the harmonic series gives divergence.
  2. The verified result is diverges\text{diverges}.

What is included

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

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