Telescoping Series

Calculus 2 · Axiom Academy

When a series is written as a difference of consecutive terms, neighbors cancel and the sum collapses like a telescope — leaving just the first term. A telescoping series is one whose terms are a difference of consecutive quantities . Write out a partial sum and almost everything cancels with its neighbor. The classic example: 2. Partial Fractions Make It Telescope Most series don't start out as a visible difference. The engine that turns a rational term into one is partial fraction decomposition . Take the term and split it. Solve by multiplying through by n(n+1) , giving 1 = A(n+1) + Bn . Matching coefficients forces A = 1 and B = -1 : The same trick, more generally Any time a denominator factors into two pieces that differ by a constant — n(n+1) , (2n-1)(2n+1) , n(n+2) — partial fractions rewrite the term as a difference. That difference is what lets the sum collapse. 3. The Partial Sum and Its Limit After cancellation the partial sum S_n keeps only the very first term and the very last negative tail: The whole sum is the limit of S_n . As the tail , so the partial sums climb toward 1 and never overshoot it: You've seen why a telescoping series collapses: written as a difference of consecutive terms, the interior cancels and only the endpoints decide the sum. Scroll up to revisit any step.

This is the written version of the interactive lesson above. See the full Calculus 2 course.