Maths

Sum

Given a sequence $a_1, a_2...a_n$ of numbers, where n is a nonnegative integer, we can write the finite sum $a_1 + a_2 + ... + a_n$ as $$\sum_{k=1}^n a_k$$ Given an infinite sequence $a_1, a_2...$ of numbers, we can write the infinite sum $a_1 + a_2 + ...$ as $$\sum_{k=1}^\infty a_k$$ which we interpret to mean $$\lim_{n\to\infty}\sum_{k=1}^n a_k$$ If the limit does not exist, the series diverges（发散）; otherwise, it converges（收敛）. The terms of a convergent series cannot always be added in any order. We can, however, rearrange the terms of an absolutely convergent series, that is, a series $\sum_{k=1}^\infty a_k$ for which the series $\sum_{k=1}^\infty |a_k|$ also converges.