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Sequences and Series

Quick questions on Method of differences 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 are producing the difference with partial fractions?
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A term like 1r(r+1)\dfrac{1}{r(r+1)} does not look like a difference, but partial fractions reveals one:
What are wider gaps?
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If the difference is f(r)−f(r+2)\mathrm{f}(r) - \mathrm{f}(r + 2) (a gap of two), then two terms survive at each end: f(1)+f(2)\mathrm{f}(1) + \mathrm{f}(2) at the start and −f(n+1)−f(n+2)-\mathrm{f}(n+1) - \mathrm{f}(n+2) at the end. Always write out the first few and last few terms to see exactly what remains.
What are sign error in partial fractions?
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Verify by recombining the partial fractions back to the original before summing.
What is q1?
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Given 1r−1r+1=1r(r+1)\dfrac{1}{r} - \dfrac{1}{r+1} = \dfrac{1}{r(r+1)}, find ∑r=1n1r(r+1)\displaystyle\sum_{r=1}^{n} \dfrac{1}{r(r+1)}. [3 marks]
What is q2?
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State what is meant by a telescoping sum and why it simplifies. [2 marks]
What is q3?
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Deduce the sum to infinity of ∑r=1∞1r(r+1)\displaystyle\sum_{r=1}^{\infty} \dfrac{1}{r(r+1)}. [2 marks]

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