Telescoping Sums: From Σ 1/(k(k+1)) to a General Method

We evaluate the finite sum 1/(1·2) + 1/(2·3) + ... + 1/(100·101) = 100/101 in three complementary ways: a direct proof by partial fractions and telescoping, a proof by mathematical induction, and two geometric arguments that make the cancellation visible. We then isolate the underlying mechanism, a discrete analogue of the Fundamental Theorem of Calculus, and use it to evaluate several families of related series: reciprocals of products of terms in arithmetic progression, reciprocals of products of r consecutive integers, factorial and trigonometric telescopes, and a short proof that Σ 1/k² < 2. We close by discussing when the method applies and when it does not, and with two open problems at its boundary: the irrationality of Euler's constant γ and the Erdős–Straus conjecture on Egyptian fractions.