infinite series

An infinite series is what you get when you try to add up infinitely many numbers, one after another, forever. It sounds impossible to finish, yet sometimes the running total creeps closer and closer to a fixed amount. Walk half the way to a wall, then half of what remains, then half again — you never arrive in one step, but the distances you cover add up to the whole journey.

Written out, a series is sum a_n = a_1 + a_2 + a_3 + ..., built from a sequence of terms a_n. To give it a value we look at the partial sums: S_1 = a_1, S_2 = a_1 + a_2, S_3 = a_1 + a_2 + a_3, and so on. These partial sums form their own sequence. If S_n approaches a finite limit S as n -> infinity, we say the series converges and its sum is S. So the value of an infinite sum is defined as the limit of its partial sums, not by literally adding forever.

Adding infinitely many positive numbers can still give a finite total, which surprises people. The half-the-distance series 1/2 + 1/4 + 1/8 + ... adds up to exactly 1. But it can also fail: 1 + 1/2 + 1/3 + 1/4 + ... grows without bound even though its terms shrink to 0, so convergence always has to be checked, never assumed.