Question

Let \( a_n = \dfrac{(-1)^{n}}{\sqrt{1+n}} \) and let \( c_n = \sum_{k=0}^{n} a_{n-k} a_k \), where \( n \in \mathbb{N} \cup \{0\} \). Then which one of the following is TRUE?
 

• A. Both \( \sum_{n=0}^{\infty} a_n \) and \( \sum_{n=1}^{\infty} c_n \) are convergent
• B. \( \sum_{n=0}^{\infty} a_n \) is convergent but \( \sum_{n=1}^{\infty} c_n \) is not convergent
• C. \( \sum_{n=1}^{\infty} c_n \) is convergent but \( \sum_{n=0}^{\infty} a_n \) is not convergent
• D. Neither \( \sum_{n=0}^{\infty} a_n \) nor \( \sum_{n=1}^{\infty} c_n \) is convergent

Hint:

Check for alternating series and use tests such as the alternating series test for convergence. When summing products of sequences, investigate whether the terms tend to 0 and their behavior.

Answer 1

The Correct Option is B

Step 1: Analyzing the series.
The sequence \( a_n = \frac{(-1)^n}{\sqrt{1+n}} \) is alternating and tends to 0 as \( n \to \infty \), suggesting that \( \sum_{n=0}^{\infty} a_n \) may be convergent. We check for the convergence of \( \sum_{n=0}^{\infty} a_n \) using the alternating series test.

Step 2: Understanding \( c_n \).
The sequence \( c_n = \sum_{k=0}^{n} a_{n-k} a_k \) involves summing products of terms from \( a_n \), and it can be shown that \( c_n \) does not converge. Thus, \( \sum_{n=1}^{\infty} c_n \) is not convergent.

Step 3: Conclusion.
The correct answer is (B) \( \sum_{n=0}^{\infty} a_n \) is convergent but \( \sum_{n=1}^{\infty} c_n \) is not convergent.