Summation of series and the method of differences

Learning objectives

Recall and apply the standard results Σr = ½n(n + 1), Σr² = ⅙n(n + 1)(2n + 1) and Σr³ = ¼n²(n + 1)², and combine them to sum any polynomial in r

Sum a series by the method of differences by writing the general term in the form f(r) − f(r + 1), listing the first and last few terms explicitly and cancelling the matching pairs

Split a rational general term into partial fractions in order to put it into a difference form, and identify how many terms fail to cancel at each end

Find the sum of a series whose limits are not 1 to n, for example from n + 1 to 2n, by subtracting one partial sum from another

Decide whether a series converges by taking the limit of its partial sum as n → ∞, state the sum to infinity when the limit exists, and explain why no sum exists when the partial sum is unbounded