Term describing the sum of a sequence where terms in the middle of the sum cancel out so that only the first and last terms remain. For example, if we take a sequence a1, a2, a3, ... and
form another sequence by taking successive differences of the form (a1 - a2), (a2 - a3), (a3 - a4), etc., then the sum of n terms of this second sequence is a telescoping sum:
a1 - a2 + a2 - a3 + a3 - a4 + ... + an - an+1 = a1 - an+1
Whilst this looks trivial, the idea is a useful one in calculating more complex
summations. For example, it is not immediately obvious how to calculate the sum of the first n terms of a sequence such as 1 / k(k+1), the unit fractions
with denominators 2, 6, 12, etc. The trick is to express this as
a telescoping sum using the fact that 1 / k(k+1) = 1 / k - 1 / (k+1), which immediately gives the sum of n terms of this sequence as 1 - 1 / (n+1) = n / (n+1). |
