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

Step 1: Understanding the Concept:
An infinite series \(\sum a_n\) is said to be absolutely convergent if the series of its absolute values, \(\sum |a_n|\), is convergent.
If \(\sum |a_n|\) converges, then the original series \(\sum a_n\) is also guaranteed to converge.

Step 3: Detailed Explanation:
Let us write down the given terms of the series:
\[ a_n = (-1)^n \frac{\sin nx}{n^2} \]
We first analyze the absolute value of the general term \(a_n\) to test for absolute convergence:
\[ |a_n| = \left| (-1)^n \frac{\sin nx}{n^2} \right| = \frac{|\sin nx|}{n^2} \]
We know that the sine function is bounded for all real numbers \(x\) and all positive integers \(n\):
\[ |\sin nx| \le 1, \quad \forall x \in \mathbb{R}, n \in \mathbb{N} \]
Applying this inequality, we obtain a bound for our absolute terms:
\[ |a_n| \le \frac{1}{n^2} \]
Now, let us consider the auxiliary series:
\[ \sum_{n=1}^\infty b_n = \sum_{n=1}^\infty \frac{1}{n^2} \]
This auxiliary series is a standard p-series of the form \(\sum \frac{1}{n^p}\), where \(p = 2\).
According to the p-series test, a series of this form converges if and only if \(p > 1\).
Since \(p = 2 > 1\), the series \(\sum_{n=1}^\infty \frac{1}{n^2}\) is convergent.
By the Direct Comparison Test, since \(0 \le |a_n| \le b_n\) for all \(n\), and \(\sum b_n\) converges, the series of absolute values \(\sum_{n=1}^\infty |a_n|\) must also converge.
Since \(\sum_{n=1}^\infty |a_n|\) converges, the original series converges absolutely.

Step 4: Final Answer:
Therefore, the correct option is (A).