Absolute Convergence and Divergence — Question 3

*Question 3 Consider $\displaystyle\sum_{n=1}^{\infty}\frac{\cos n}{n^2}$.

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.

*Question 3 – Solution

Since $|\cos n|\le1$, $$0\le\left|\frac{\cos n}{n^2}\right|\le\frac1{n^2}.$$

The benchmark $\sum1/n^2$ converges. Direct Comparison gives $\sum|\cos n|/n^2<\infty$.

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