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Mathematical Induction, Inequalities and Recurrences

Quick questions on Summation of series explained: H2 Further Mathematics

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 the method of differences (telescoping)?
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If a term can be written as a difference ur=f(r)−f(r+1)u_r = \mathrm{f}(r) - \mathrm{f}(r+1), then the sum collapses:
What is using partial fractions to telescope?
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Many fractional terms become telescoping after partial fractions. For example 1r(r+1)=1r−1r+1\dfrac{1}{r(r+1)} = \dfrac{1}{r} - \dfrac{1}{r+1} has exactly the difference form. For 1r(r+2)\dfrac{1}{r(r+2)} the gap is two, so the cancellation leaves the first two and last two terms.
What is not displaying the cancellation?
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The proof structure (write the first and last brackets) is itself worth marks.
What is q1?
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Write 1r(r+1)\dfrac{1}{r(r+1)} as a difference of two fractions. [1 mark]
What is q2?
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Use the standard results to find ∑r=1n(r2+r)\sum_{r=1}^{n} (r^2 + r). [2 marks]
What is q3?
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State the sum to infinity of ∑r=1∞(1r−1r+1)\sum_{r=1}^{\infty}\left(\dfrac{1}{r} - \dfrac{1}{r+1}\right). [1 mark]

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