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Questions › IIT JAM › IIT JAM 2021 › Q9
1 mark, −0.333Mathematics › Sequences, series and convergence › Convergence of seriesModerateabout 1 min

IIT JAM 2021 Question 9: Sequences, series and convergence

Which one of the following statements is correct?

Tap an option to check your answer.

Answer and worked solution

Answer: (D)

A necessary condition for ∑an\sum a_n∑an​ to converge is that an→0a_n \to 0an​→0, so statement (D) is correct.

The others fail: a bounded sequence need not converge (for example (−1)n(-1)^n(−1)n), a convergent sequence need not be monotonic, and an→0a_n \to 0an​→0 does not guarantee the series converges (the harmonic series diverges). The answer is (D).

The trap: (C) If ⟨an⟩\langle a_n \rangle⟨an​⟩ converges to zero, then the series ∑n=1∞an\sum_{n=1}^{\infty} a_n∑n=1∞​an​ is convergent

Statement (C) reverses the implication: terms going to zero is necessary but not sufficient, as the divergent harmonic series shows.

Plain-text version

Which one of the following statements is correct? (A) If ⟨ aₙ ⟩ is a bounded sequence, then it is convergent (B) If ⟨ aₙ ⟩ is a convergent sequence, then it is monotonic (C) If ⟨ aₙ ⟩ converges to zero, then the series Σₙ₌₁^∞ aₙ is convergent (D) If a series Σₙ₌₁^∞ aₙ is convergent, then the sequence ⟨ aₙ ⟩ converges to zero

Sequences, series and convergence has come up 33 times in 5 exams since 2006.

What this question tests

This question is on Convergence of series, part of Sequences, series and convergence in Mathematics.

Tests for whether an infinite sum converges.

TestStop has 11 past-paper questions on this subtopic.

Take this paper as a timed mockMore on Convergence of series

Question from the IIT JAM 2021 paper. Solutions are TestStop's own. Not affiliated with or endorsed by any examination authority.