Sequences and Infinite Sums
An infinite sum is really just a limit of partial sums, and that single idea explains when adding forever can still land on a finite number.
Sequence limits
A sequence is an infinite list of numbers written as a_n, and it converges to L if the terms get arbitrarily close to L as n grows large. For example, a_n = 1/n approaches 0, so the sequence converges to 0. If the terms grow without bound or oscillate, the sequence diverges, and any series built from its terms diverges too.
a_n = \frac{1}{n} \to 0From list to sum
A series is what you get when you add the terms of a sequence, written as the sum of a_n from n equal 1 to infinity. The key object is the nth partial sum s_n, an ordinary finite sum of the first n terms. The series converges to S exactly when the sequence of partial sums converges to S.
s_n = \sum_{k=1}^{n} a_kPartial sums example
Take the series 1/2 + 1/4 + 1/8 + dots, where each term is half the one before it. The partial sums run 1/2, 3/4, 7/8, and so on, each step closing in on 1. Since s_n = 1 - (1/2)^n, the partial sums approach 1, so the infinite sum equals exactly 1. You can halve the remaining gap forever, and the gap shrinks to nothing.
Flashcard
The sequence converges to the limit L
Check your understanding
What is the value of the infinite series 1/2 + 1/4 + 1/8 + dots, where each term is half the previous one?