Series

• If u_r = f(r) − f(r + 1), then Σ from r = 1 to n of u_r = f(1) − f(n + 1).
• Write out the first few and last few terms in columns to see exactly which terms survive.
• Partial fractions turn terms like 1/(r(r + 1)) into differences.
• When the "gap" is 2, as in 1/(r(r + 2)), two terms survive at each end.
• Sums from n + 1 to 2n: use S(2n) − S(n).

Watching the terms cancel

Write u_r = f(r) − f(r + 1) and list the terms:

r = 1: f(1) − f(2)
r = 2: f(2) − f(3)
⋮
r = n: f(n) − f(n + 1)

Adding, every f(2), f(3), … f(n) cancels, leaving f(1) − f(n + 1). Always show this listing in an exam — the cancellation is where the method marks are.

Making the difference appear

Express the general term in partial fractions first. Then list terms and look for the cancelling pattern. With a gap of 1, one term survives at each end; with a gap of 2, two terms survive at each end.