Summation of series — A/L Combined Mathematics

What must I remember about Summation of series?

• Method of Differences (Telescoping Sums): Express the general term ur as the difference of two consecutive terms of another sequence: ur = f(r) - f(r + 1) or ur = f(r + 1) - f(r).
• Telescoping cancellation: In ∑_r=1ⁿ [f(r) - f(r + 1)] = [f(1) - f(2)] + [f(2) - f(3)] + ... + [f(n) - f(n + 1)], all intermediate terms cancel pairwise, leaving Sn = f(1) - f(n + 1).
• Partial Fractions decomposition for telescoping: For ur = 1 / [r(r + 1)], write ur = 1/r - 1/(r + 1); summing from 1 to n gives Sn = 1 - 1/(n + 1) = n/(n + 1).
• Factor method for product series: For ur = r(r + 1)(r + 2), write ur = (1/4)[r(r + 1)(r + 2)(r + 3) - (r - 1)r(r + 1)(r + 2)]; summing yields Sn = (1/4)n(n + 1)(n + 2)(n + 3).
• Arithmetico-Geometric Series (A.G.P.): Series of the form S = a + (a + d)r + (a + 2d)r² + ...; summed by multiplying S by common ratio r and subtracting: S - rS = (1 - r)S.

Test yourself (1 minute)

1. Identifying f(r) in reciprocal products

   For ur = 1 / [(ar + b)(a(r+1) + b)], let f(r) = 1 / (ar + b); then f(r) - f(r + 1) = a / [(ar + b)(a(r+1) + b)] = a * ur, so ur = (1/a)[f(r) - f(r + 1)].

2. Split cancellation over two steps

   If ur = f(r) - f(r + 2), the sum retains TWO initial terms and TWO terminal terms: Sn = f(1) + f(2) - f(n + 1) - f(n + 2).

3. Infinite sum from telescoping

   If lim_n->∞ f(n + 1) = 0, the sum to infinity exists and equals S_∞ = f(1).

Practise Summation of series questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 20

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Where does Summation of series fit in the syllabus?

Year 1 · Term 3