Telescoping Sums

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

1/(1x2) + 1/(2x3) + ... + 1/(NxN+1) = N/(N+1)

Split every term by partial fractions: 1/(k(k+1)) is 1/k − 1/(k+1).

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

Only the outermost terms are left: 1 − 1/(N + 1), which is N/(N + 1).

Your turn

Three to try in your head — tap what you get.

Telescoping sum to 1/(12×13)

Telescoping sum to 1/(6×7)

Telescoping sum to 1/(10×11)