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 a_{1}, a_{2}, a_{3}, ... and

form another sequence by taking successive differences of the form (a_{1} - a_{2}), (a_{2} - a_{3}), (a_{3} - a_{4}), etc., then the sum of n terms of this second sequence is a telescoping sum:

a_{1} - a_{2} + a_{2} - a_{3} + a_{3} - a_{4} + ... + a_{n} - a_{n+1} = a_{1} - a_{n+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).