Summation of series

Further Maths · Further Pure Mathematics 1. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

This topic is about adding the first n terms of a series and getting a formula in n. There are two main tools.

The first is the three standard results for Σr, Σr² and Σr³. Split the general term into powers of r, sum each part, then factorise.

The second is the method of differences. If each term can be written as f(r) − f(r + 1), often by using partial fractions, then almost all the terms cancel and only a few at the start and the end are left.

A series is convergent if the sum to n terms tends to a finite limit as n → ∞. That limit is the sum to infinity.

Rules to remember

Σ_r=1ⁿ r = n(n + 1)/2
Σ_r=1ⁿ r² = n(n + 1)(2n + 1)/6
Σ_r=1ⁿ r³ = n²(n + 1)²/4
Σ_r=1ⁿ 1 = n, so a constant c sums to cn, not c
Sum from r = m + 1 to n = (sum from 1 to n) − (sum from 1 to m)
Method of differences: if u_r = f(r) − f(r + 1), then Σ_r=1ⁿ u_r = f(1) − f(n + 1)
Sum to infinity = limit of S_n as n → ∞, if this limit is finite; terms such as 1/n and 1/(n + 2) tend to 0

Common mistake

In the method of differences students cancel too quickly and keep the wrong terms. Write out the first two or three lines and the last two lines in full, then see which terms have no partner.

Worked example

Find Σ_r=1ⁿ 1/((r + 1)(r + 2)) and state the sum to infinity.

Partial fractions: 1/((r + 1)(r + 2)) = 1/(r + 1) − 1/(r + 2).
Write out the terms: (1/2 − 1/3) + (1/3 − 1/4) + … + (1/(n + 1) − 1/(n + 2)).
Everything cancels except the first and the last: sum = 1/2 − 1/(n + 2). Check n = 1: 1/2 − 1/3 = 1/6, which is 1/(2 × 3).
As n → ∞, 1/(n + 2) → 0, so the sum to infinity is 1/2.

Answer: Sum to n terms = 1/2 − 1/(n + 2); sum to infinity = 1/2

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