Question

Which one of the following statements is correct?

• A. If $\langle a_n \rangle$ is a bounded sequence, then it is convergent
• B. If $\langle a_n \rangle$ is a convergent sequence, then it is monotonic
• C. If $\langle a_n \rangle$ is a convergent sequence and converges to zero, then the series $\sum_{n=1}^{\infty} a_n$ is convergent
• D. If a series $\sum_{n=1}^{\infty} a_n$ is convergent, then the sequence $\langle a_n \rangle$ is convergent and converges to zero

Hint:

A convergent series must have terms tending to zero, but the reverse is not always true — this is a common trap in sequence and series questions.

Answer 1

The Correct Option is D

Step 1: Recall property of convergence.
If $\sum a_n$ converges, the necessary condition is that $a_n \to 0$ as $n \to \infty$.
Step 2: Analyze options.
(A) False — a bounded sequence need not converge (e.g., $\sin n$).
(B) False — convergence does not imply monotonicity.
(C) False — even if $a_n \to 0$, the series can diverge (e.g., harmonic series).
(D) True — convergence of series implies the sequence terms tend to zero.

Step 3: Conclusion.
Thus, the correct statement is (D).