How a Telescoping Sum Collapses to Its Two Ends

Updated 4 October 2026

Rewrite each term so almost everything cancels — only the ends survive.

$\frac{1}{1x2}+\frac{1}{2x3}+\cdots +\frac{1}{\text{NxN}+1}$1/(1x2) + 1/(2x3) + ... + 1/(NxN+1) = $\frac{N}{N+1}$N/(N+1)

How does a telescoping sum collapse to its two ends?

A telescoping sum collapses because each term splits into two pieces that cancel with its neighbors. For 1/(1 × 2) + 1/(2 × 3) + … + 1/(N(N + 1)), split each term by partial fractions: 1/(k(k + 1)) is 1/k − 1/(k + 1). Written out, the sum becomes (1 − 1/2) + (1/2 − 1/3) + (1/3 − 1/4) and so on. Each minus 1/2 meets a plus 1/2, each minus 1/3 meets a plus 1/3, and every inside pair cancels. Only the first piece of the first term and the last piece of the last term survive, so the sum is 1 − 1/(N + 1), which is N/(N + 1). For N = 4, the sum is 1/2 + 1/6 + 1/12 + 1/20, which the formula gives as 4/5. Check it with a common denominator of 60: 30 + 10 + 5 + 3 = 48, and 48/60 is 4/5. As N grows, the sum gets closer to 1 without reaching it.

1. Split every term by partial fractions: $\frac{1}{k\left(k+1\right)}$1/(k(k+1)) is $\frac{1}{k}-\frac{1}{k+1}$1/k − 1/(k+1).

2. Written out, each inside pair cancels: −½ meets +½, and −⅓ meets +⅓.

3. Only the outermost terms are left: $1-\frac{1}{N+1}$1 − 1/(N + 1), which is $\frac{N}{N+1}$N/(N + 1).

All 36 shortcuts: the Magic Tricks library.