Question

Which of the following statements are correct:
A. If $\sum a_n$ converges, then $\lim_{n \to \infty} a_n = 0$ B. If $|a_n| \le c_n$ for all $n$, and if $\sum c_n$ diverges, then $\sum a_n$ diverges.
C. If $|a_n| \le c_n$ for $n \ge N_0$, ($N_0$ is fixed integer) and if $\sum c_n$ converges, then $\sum a_n$ converges.
D. If $a_n \ge d_n \ge 0$ for $n$, and if $\sum d_n$ converges, then $\sum a_n$ converges.
E. If $a_n \ge d_n \ge 0$ for $n \ge N_0$ and if $\sum d_n$ diverges, then $\sum a_n$ diverges.
Choose the correct answer from the options given below:

• A. A, B, D Only
• B. A, B, E Only
• C. A, C, E Only
• D. A, C, D Only

Hint:

Remember the Comparison Test direction: - Bigger series converges $\implies$ Smaller series converges. - Smaller series diverges $\implies$ Bigger series diverges.

Answer 1

The Correct Option is C

Step 1: Concept:
This question tests fundamental convergence tests for infinite series, specifically the $n$-th term test and the Comparison Test.

Step 2: Key Formula or Approach:
1. $n$-th term test for convergence: $\sum a_n \text{ converges} \implies \lim_{n \to \infty} a_n = 0$.
2. Comparison Test: Let $0 \le a_n \le c_n$ for $n \ge N_0$.
- If $\sum c_n$ converges, then $\sum a_n$ converges (Direct Comparison Test for Convergence).
- If $\sum a_n$ diverges, then $\sum c_n$ diverges.

Step 3: Step-by-step Explanation:

• Statement A:
It is a necessary condition for convergence that the sequence of terms $a_n$ approaches zero as $n \to \infty$. Statement A is correct.

• Statement B:
If $|a_n| \le c_n$ and $\sum c_n$ diverges, no conclusion can be drawn about $\sum a_n$ (e.g., $a_n = \frac{1}{n^2}$, $c_n = \frac{1}{n}$; $\sum c_n$ diverges but $\sum a_n$ converges). Statement B is false.

• Statement C:
If $|a_n| \le c_n$ and $\sum c_n$ converges, then $\sum |a_n|$ converges by comparison. Absolute convergence implies convergence. Statement C is correct.

• Statement D:
If $a_n \ge d_n \ge 0$ and $\sum d_n$ converges, $a_n$ is larger than $d_n$, so $\sum a_n$ might diverge (e.g., $d_n = \frac{1}{n^2}$, $a_n = \frac{1}{n}$). Statement D is false.

• Statement E:
If $a_n \ge d_n \ge 0$ and the smaller series $\sum d_n$ diverges, then the larger series $\sum a_n$ must also diverge to infinity. Statement E is correct.

Step 4: Final Answer:
Statements A, C, and E are correct. Therefore, option (C) is the correct answer.