information:extraction, linguistic:translation, structural:summarization, structural:reformatting
Background

When adding a finite number of terms, addition is both associative and commutative, meaning that grouping and rearrangment do not alter the final sum. For instance, (1 + 2) + 3 {\displaystyle (1+2)+3} is equal to both 1 + (2 + 3) {\displaystyle 1+(2+3)} and (3 + 2) + 1 {\displaystyle (3+2)+1}. However, associativity and commutativity do not necessarily hold for infinite sums. One example is the alternating harmonic series

S = ∑ n = 1 ∞ (− 1) n + 1 n = 1 − 1 2 + 1 3 − 1 4 + 1 5 − 1 6 + ⋯ {\displaystyle S=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}=1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-{\frac {1}{6}}+\cdots }

whose terms are fractions that alternate in sign. This series is convergent and can be evaluated using the Maclaurin series for the function ln (1 + x) {\displaystyle \ln(1+x)}, which converges for all x {\displaystyle x} satisfying − 1 < x ≤ 1 {\displaystyle -1<x\leq 1}:

ln (1 + x) = ∑ n = 1 ∞ (− 1) n + 1 x n n = x − x 2 2 + x 3 3 − x 4 4 + ⋯ {\displaystyle \ln(1+x)=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}x^{n}}{n}}=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}+\cdots }

Substituting x = 1 {\displaystyle x=1} reveals that the original sum is equal to ln 2 {\displaystyle \ln 2}. The sum can also be rearranged as follows:

S = (1 − 1 2) − 1 4 + (1 3 − 1 6) − 1 8 + (1 5 − 1 10) − 1 12 + ⋯ {\displaystyle S=\left(1-{\frac {1}{2}}\right)-{\frac {1}{4}}+\left({\frac {1}{3}}-{\frac {1}{6}}\right)-{\frac {1}{8}}+\left({\frac {1}{5}}-{\frac {1}{10}}\right)-{\frac {1}{12}}+\cdots }

In this rearrangement, the reciprocal of each odd number is grouped with the reciprocal of twice its value, while the reciprocals of every multiple of 4 are evaluated separately. However, evaluating the terms inside the parentheses yields

S = 1 2 − 1 4 + 1 6 − 1 8 + 1 10 − 1 12 + ⋯ {\displaystyle S={\frac {1}{2}}-{\frac {1}{4}}+{\frac {1}{6}}-{\frac {1}{8}}+{\frac {1}{10}}-{\frac {1}{12}}+\cdots }

or half the original series. The violation of the associativity and commutativity of addition reveals that the alternating harmonic series is conditionally convergent. Indeed, the sum of the absolute values of each term is 1 + 1 2 + 1 3 + 1 4 + ⋯ {\textstyle 1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+\cdots }, or the divergent harmonic series. According to the Riemann series theorem, any conditionally convergent series can be permuted so that its sum is any finite real number or so that it diverges. When an absolutely convergent series is rearranged, its sum is always preserved.