Question

Let \(\sum_{n=1}^{\infty}a_n\) be any series of real numbers, then which of the following statement is true?

• A. If $\sum_{\text{n}=1}^{\infty} \text{a}_\text{n}$ is absolutely convergent then it is convergent
• B. If $\sum_{\text{n}=1}^{\infty} \text{a}_\text{n}$ is convergent then it is absolutely convergent
• C. If $\sum_{\text{n}=1}^{\infty} \text{a}_\text{n}$ is absolutely convergent then it is conditionally convergent
• D. If $\sum_{\text{n}=1}^{\infty} \text{a}_\text{n}$ is conditionally convergent then it is absolutely convergent

Hint:

Absolute Convergence $\implies$ Ordinary Convergence.
Ordinary Convergence $\kern-0.3em\not\kern-0.3em\implies$ Absolute Convergence (as shown by conditionally convergent series like \( \sum \frac{(-1)^n}{n} \)).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We evaluate the relationship between absolute convergence, conditional convergence, and ordinary convergence of infinite series.

Step 2: Detailed Explanation:
Let us review the standard mathematical definitions and theorems for infinite series:
1. Absolute Convergence:
A series \( \sum a_n \) is absolutely convergent if the series of absolute values \( \sum |a_n| \) is convergent.
2. Absolute Convergence Theorem:
A fundamental theorem in real analysis states that: "Absolute convergence implies convergence."
If \( \sum |a_n| \) converges, then the original series \( \sum a_n \) must also converge.
This confirms that Option A is a true statement.
3. Conditional Convergence:
A series \( \sum a_n \) is conditionally convergent if \( \sum a_n \) converges but \( \sum |a_n| \) diverges (for example, the alternating harmonic series \( \sum \frac{(-1)^{n+1}}{n} \)).
This definition shows that absolute convergence and conditional convergence are mutually exclusive, which makes Option C and Option D false.
Since there are many conditionally convergent series that are not absolutely convergent, ordinary convergence does not imply absolute convergence, making Option B false.

Step 3: Final Answer:
The true statement is: If $\sum_{\text{n}=1}^{\infty} \text{a}_\text{n}$ is absolutely convergent then it is convergent (Option A).