Partial Sums and Telescoping Series

To add infinitely many numbers, you never actually add infinitely many numbers at once. You build partial sums: S₁ = a₁, S₂ = a₁ + a₂, S₃ = a₁ + a₂ + a₃, and so on. The infinite series is defined to converge if this sequence of partial sums approaches a single number L as n goes to infinity. That limit L is the sum. If the partial sums grow without bound or bounce around forever, the series diverges and has no sum.

Telescoping series are the cleanest case because you can find a formula for S_n directly instead of guessing a pattern. The trick is rewriting each term as a difference of two pieces that overlap with neighboring terms, like 1/(n(n+1)) = 1/n - 1/(n+1), using partial fractions. When you list out the partial sum, every interior term appears once with a plus sign and once with a minus sign, so it cancels. What's left is just the first leftover piece and the last leftover piece. Taking the limit as n goes to infinity often sends the last piece to zero, leaving a clean finite answer.

This topic is the foundation for everything else in the unit: every convergence test you learn later is really a way of predicting the behavior of partial sums without being able to compute them exactly, the way telescoping lets you here. Geometric series, which come next, are the other major family where S_n has a closed form you can take a limit of directly.