section_path: Examples of numerical series/Natural logarithm of 2
section_name: Natural logarithm of 2
paragraph_text: Natural logarithm of 2
  
  ∑ n = 1 ∞ (− 1) n + 1 n = ln 2 {\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}=\ln 2}
  
  ∑ n = 0 ∞ 1 (2 n + 1) (2 n + 2) = ln 2 {\displaystyle \sum _{n=0}^{\infty }{\frac {1}{(2n+1)(2n+2)}}=\ln 2}
  
  ∑ n = 0 ∞ (− 1) n (n + 1) (n + 2) = 2 ln (2) − 1 {\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(n+1)(n+2)}}=2\ln(2)-1}
  
  ∑ n = 1 ∞ 1 n (4 n 2 − 1) = 2 ln (2) − 1 {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n\left(4n^{2}-1\right)}}=2\ln(2)-1}
  
  ∑ n = 1 ∞ 1 2 n n = ln 2 {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{2^{n}n}}=\ln 2}
  
  ∑ n = 1 ∞ (1 3 n + 1 4 n) 1 n = ln 2 {\displaystyle \sum _{n=1}^{\infty }\left({\frac {1}{3^{n}}}+{\frac {1}{4^{n}}}\right){\frac {1}{n}}=\ln 2}
  
  ∑ n = 1 ∞ 1 2 n (2 n − 1) = ln 2 {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{2n(2n-1)}}=\ln 2}