Convergence of a sequence
Finite or infinite limit of a sequence, reference sequences, comparison and squeeze theorems, bounded monotone sequences, behaviour of a geometric sequence according to its ratio, and finding a threshold.
- 25 cards
- 6 topics
- Free
The outline, topic by topic
Each topic is weighted the way the exam weighs it.
- Concepts
- Questions
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A few cards from the deck
Limit of a sequence: what is it?
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A value ℓ that the terms approach as closely as you wish, and stay close to, from some index onwards.
If a sequence has a limit, it is unique. The terms need not reach ℓ: 1/n tends to 0 without ever equalling 0.
What is the limit of (3n + 1)/(2n − 5) as n tends to +∞?
- 3/2
- 3
- 0
- +∞
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3/2
Factor out the dominant term: (3n + 1)/(2n − 5) = (3 + 1/n)/(2 − 5/n), which tends to 3/2. The 1/n and 5/n terms vanish, but neither numerator nor denominator has a finite limit on its own.
The sequence qⁿ tends to … if …, equals 1 for q = 1, tends to +∞ if q > 1 and has no limit if q ≤ −1.
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The sequence qⁿ tends to 0 if |q| < 1, equals 1 for q = 1, tends to +∞ if q > 1 and has no limit if q ≤ −1.
For −1 < q < 0, the terms alternate in sign but their absolute value decreases to 0. For q = −1, they are alternately 1 and −1: no limit.
Convergent sequence: what is it?
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A sequence whose limit is a real number.
A sequence that tends to +∞ does not converge: infinity is not a real number. Neither does (−1)ⁿ, which has no limit. It is said to diverge in both cases.