Partial fractions in series

Objective. Decompose terms to create telescoping behaviour.

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Learn: the key idea

Partial fractions can decompose a complicated rational term into simpler fractions that create telescoping cancellation when summed. A term like 1/[(r+1)(r+2)] is rewritten as 1/(r+1) − 1/(r+2) using partial fraction decomposition, after which the method of differences applies directly. This combines two skills: finding the partial fraction form algebraically, then summing the resulting telescoping series over the given range. For example, decomposing 2/[r(r+2)] gives 1/r − 1/(r+2), which telescopes with a two-term gap rather than adjacent cancellation.

Worked example

Express 1/[r(r+1)(r+2)] in partial fractions and use this to find Σ 1/[r(r+1)(r+2)] for r=1 to n.

1 $Using partial fractions, 1/[r(r+1)(r+2)] = (1/2)[1/(r(r+1)) − 1/((r+1)(r+2))]$
2 $Apply the method of differences: the sum telescopes to (1/2)[1/(1\times2) - 1/((n+1)(n+2))]$
3 $Simplify the first bracket term: 1/2 \times 1/2 = 1/4$
4 $So the sum equals 1/4 - 1/[2(n+1)(n+2)]$

Answer: 1/4 − 1/[2(n+1)(n+2)]

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Common mistake. Students forget that with a two-term gap like 1/r − 1/(r+2), two terms survive at each end rather than one, since consecutive terms don't fully cancel. Always list enough terms to see exactly which survive.

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