Summation of series — A/L Combined Mathematics

The Method of Differences (telescoping series) is a powerful algebraic technique used to sum series where standard power formulas do not apply. The r-th term is expressed as the difference of consecutive values of a function: u_r = f(r) - f(r + 1) or u_r = f(r + 1) - f(r). When terms are summed from r = 1 to n, intermediate terms cancel cyclically: ∑_r=1ⁿ u_r = [f(1) - f(2)] + [f(2) - f(3)] + ... + [f(n) - f(n + 1)] = f(1) - f(n + 1). For rational fraction terms, resolving u_r into partial fractions is the primary method to achieve this cancellation.

Check before you answer (A/L Pure Maths Paper 1)

1. Partial Fractions Decomposition: Resolving rational r-th terms accurately into differences of partial fractions (e.g., 1 / [r(r + 1)] = 1/r - 1/(r + 1)).
2. Explicit Listing of Boundary Terms: Writing down the first two terms (r = 1, 2) and the last two terms (r = n - 1, n) to clearly display the cancellation pattern.
3. Sum to Infinity Deduction: Evaluating the limit lim_n→∞ S_n directly from the simplified S_n expression.