Question

Let \[ \sum_{n=1}^{\infty} a_n = \sum_{n=1}^{\infty} (-1)^n \frac{\sin(nx)}{n^2} \] be a series. Then: 

• A. \(\sum_{n=1}^\infty a_n\) convergent absolutely
• B. \(\sum_{n=1}^\infty a_n\) convergent conditionally
• C. \(\sum_{n=1}^\infty a_n\) oscillatory
• D. \(\sum_{n=1}^\infty a_n\) is not convergent

Hint:

Whenever you see a series involving \(\frac{\sin nx}{n^p}\) or \(\frac{\cos nx}{n^p}\) with \(p > 1\), it is always absolutely convergent.
This is because both sine and cosine terms are bounded by 1, allowing for a direct comparison with a convergent p-series.

Answer 1

The Correct Option is A