STEM-Master
Calculus IIChapter 4: Series and SequencesSection 4-9

Absolute Convergence and Divergence — Question 3

PDF ↗

Question 3

Consider ∑n=1∞cos⁡nn2\displaystyle\sum_{n=1}^{\infty}\frac{\cos n}{n^2}.

  1. Bound the absolute value of each term.

  2. Apply Direct Comparison to classify absolute convergence.

  3. Explain what this implies about the original oscillatory series.

Original worksheet page 1: question and worked solution for 4-9-003
Show solutionHide solution

Question 3 – Solution

Step 1: Remove oscillation by an inequality.

Since |cos⁡n|≤1|\cos n|\le1, 0≤|cos⁡nn2|≤1n2.0\le\left|\frac{\cos n}{n^2}\right|\le\frac1{n^2}.

Step 2: Compare absolute values.

The benchmark ∑1/n2\sum1/n^2 converges. Direct Comparison gives ∑|cos⁡n|/n2<∞\sum|\cos n|/n^2<\infty.

Step 3: State the strongest conclusion.

The series converges absolutely, and therefore converges ordinarily. No cancellation or special information about the values of cos⁡n\cos n is needed.

Original worksheet page 2: question and worked solution for 4-9-003

Original worksheet layout. Use Enlarge or open the PDF for a closer view.

← Section questionsChapter contents