Here are a some sequences of numbers:a) $1, -1, 1/2, -1/2, 1/3, -1/3, \ldots$b) $1, -2, 3, -4, 5, -6, …$c) $0, 0, 0, 0, 0, …$d) $1, 2, 3, 4, 5, 6, …$e) $0+1, 1+1/2, 0+1/3, 1+1/4, 0+1/5, 1+1/6, ….$How many of these sequences are divergent?

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We are asked to determine how many of the given sequences are divergent among five labelled sequences a) through e).

Option a) a) 1, -1, 1/2, -1/2, 1/3, -1/3, ...
- The absolute values follow 1, 1, 1/2, 1/2, 1/3, 1/3, ... which tend to 0 as n grows, and the signs alternate.
- Since the terms approach 0, the sequence converges (to 0). This one is not divergent.

Option b) b) 1, -2, 3, -4, 5, -6, ...
- The absolute values are 1, 2, 3, 4, 5, 6, ... which grow without bound.
- The sequence does not approach any finite limit and is unbounded, hence divergent.

Option c) c) 0, 0, 0, 0, 0, …
- Every term is 0, so the sequence clearly converges to 0.
- This is not divergent.

Option d) d) 1, 2, 3, 4, 5, 6, …
- This is the classic unbounded increasing sequence; it does not settle to a finite limit.
- Therefore, it diverges.

Option e) e) 0+1, 1+1/2, 0+1/3, 1+1/4, 0+1/5, 1+1/6, …
- The odd-indexed terms are 1/n, which tend to 0 as n → ∞, while the even-indexed terms are 1 + 1/n, which tend to 1.
- Since the odd and even subsequences converge to different limits, the whole sequence does not converge to a single value.
- Thus this sequence is divergent.

Summary: Among the five sequences, a) converges, b) diverges, c) converges, d) diverges, e) diverges. Therefore, the number of divergent sequences is 3.

MTH1030 -1035 - S1 2025 MTH1030/5 Week 8 lesson quiz: Infinite sequences and series

Here are a some sequences of numbers:a) \(1, -1, 1/2, -1/2, 1/3, -1/3, \ldots\)b) \(1, -2, 3, -4, 5, -6, …\)c) \(0, 0, 0, 0, 0, …\)d) \(1, 2, 3, 4, 5, 6, …\)e) \(0+1, 1+1/2, 0+1/3, 1+1/4, 0+1/5, 1+1/6, ….\)How many of these sequences are divergent?

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