linguistic:paraphrasing, information:extraction
Alternating harmonic series

The series ∑ n = 1 ∞ (− 1) n + 1 n = 1 − 1 2 + 1 3 − 1 4 + 1 5 − ⋯ {\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}=1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-\cdots } is known as the alternating harmonic series. It is conditionally convergent by the alternating series test, but not absolutely convergent. Its sum is the natural logarithm of 2.

Explicitly, the asymptotic expansion of the series is

1 1 − 1 2 + ⋯ + 1 2 n − 1 − 1 2 n = H 2 n − H n = ln 2 − 1 2 n + O (n − 2) {\displaystyle {\frac {1}{1}}-{\frac {1}{2}}+\cdots +{\frac {1}{2n-1}}-{\frac {1}{2n}}=H_{2n}-H_{n}=\ln 2-{\frac {1}{2n}}+O(n^{-2})}

Using alternating signs with only odd unit fractions produces a related series, the Leibniz formula for π ∑ n = 0 ∞ (− 1) n 2 n + 1 = 1 − 1 3 + 1 5 − 1 7 + ⋯ = π 4. {\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots ={\frac {\pi }{4}}.}