Limit

The ancient Greek philosopher Zeno may have been one of the first mathematicians to ponder the limit of a sequence and wonder how it related to the world around him. Zeno argued that all motion was impossible because in order to move a distance l it is first necessary to travel half the distance, then half the remaining distance, then half of that remaining distance and so on. Thus, he argued, the distance l can never be fully traversed.

Consider the sequence 1, 1/2, 1/4, 1/8,...(1/2)ⁿ when n gets very large. Since (1/2)ⁿ equals 1/2 multiplied by itself n times, (1/2)ⁿ gets very small when n is allowed to become infinitely large. The sequence is said to converge, meaning numbers that are very far along in the sequence (corresponding to large n) get very close together and very close to a single value called the limit.

A sequence of numbers converges to a given number if the differences between the terms of the sequence and the given number form an infinitesimal sequence. For this sequence (1/2)ⁿ gets arbitrarily close to 0, so 0 is the limit of the sequence. The numbers in the sequence never quite reach the limit, but they never go past it either.

If an infinite sequence diverges, the running total of the terms eventually turns away from any specific value, so a divergent sequence has no limiting sum.