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Sequences and Series

Quick questions on Convergence and limits of sequences explained: H2 Mathematics Sequences and Series

6short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is behaviour of a sequence as n tends to infinity?
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A sequence unu_n may converge to a limit LL (the terms settle ever closer to LL), diverge to infinity, or oscillate. To find a limit of a rational expression in nn, divide numerator and denominator by the highest power of nn and use the fact that 1nk→0\dfrac{1}{n^k} \to 0.
What is convergence of a geometric series?
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The partial sums of a geometric series form their own sequence Sn=a(1−rn)1−rS_n = \dfrac{a(1 - r^n)}{1 - r}. This sequence converges exactly when rn→0r^n \to 0, which happens if and only if ∣r∣<1|r| < 1. In that case
What is interpreting a limit?
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A limit is the value the sequence (or partial sum) approaches but may never exactly reach. Saying a savings total "tends to" a figure means it gets arbitrarily close as time goes on, which is exactly the sum-to-infinity interpretation.
What is q1?
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Find the limit of un=2n2+1n2+3u_n = \dfrac{2n^2 + 1}{n^2 + 3} as n→∞n \to \infty. [2 marks]
What is q2?
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State the condition for a geometric series to converge and explain why. [2 marks]
What is q3?
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Determine whether the series with a=5a = 5, r=1.2r = 1.2 converges. [1 mark]

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