section_path: Properties of series/Conditional convergence
section_name: Conditional convergence
paragraph_text: Conditional convergence
  
  A series of real or complex numbers is said to be conditionally convergent (or semi-convergent) if it is convergent but not absolutely convergent. A famous example is the alternating series
  
  ∑ n = 1 ∞ (− 1) n + 1 n = 1 − 1 2 + 1 3 − 1 4 + 1 5 − ⋯, {\displaystyle \sum \limits _{n=1}^{\infty }{(-1)^{n+1} \over n}=1-{1 \over 2}+{1 \over 3}-{1 \over 4}+{1 \over 5}-\cdots,}
  
  which is convergent (and its sum is equal to ln 2 {\displaystyle \ln 2}), but the series formed by taking the absolute value of each term is the divergent harmonic series. The Riemann series theorem says that any conditionally convergent series can be reordered to make a divergent series, and moreover, if the a n {\displaystyle a_{n}} are real and S {\displaystyle S} is any real number, that one can find a reordering so that the reordered series converges with sum equal to S {\displaystyle S}.
  
  Abel's test is an important tool for handling semi-convergent series. If a series has the form
  
  ∑ a n = ∑ λ n b n {\displaystyle \sum a_{n}=\sum \lambda _{n}b_{n}}
  
  where the partial sums B n = b 0 + ⋯ + b n {\displaystyle B_{n}=b_{0}+\cdots +b_{n}} are bounded, λ n {\displaystyle \lambda _{n}} has bounded variation, and lim λ n b n {\displaystyle \lim \lambda _{n}b_{n}} exists:
  
  sup N | ∑ n = 0 N b n | < ∞, ∑ | λ n + 1 − λ n | < ∞ and λ n B n converges, {\displaystyle \sup _{N}\left|\sum _{n=0}^{N}b_{n}\right|<\infty,\ \ \sum \left|\lambda _{n+1}-\lambda _{n}\right|<\infty \ {\text{and}}\ \lambda _{n}B_{n}\ {\text{converges,}}}
  
  then the series ∑ a n {\textstyle \sum a_{n}} is convergent. This applies to the point-wise convergence of many trigonometric series, as in
  
  ∑ n = 2 ∞ sin (n x) ln n {\displaystyle \sum _{n=2}^{\infty }{\frac {\sin(nx)}{\ln n}}}
  
  with 0 < x < 2 π {\displaystyle 0<x<2\pi }. Abel's method consists in writing b n + 1 = B n + 1 − B n {\displaystyle b_{n+1}=B_{n+1}-B_{n}}, and in performing a transformation similar to integration by parts (called summation by parts), that relates the given series ∑ a n {\textstyle \sum a_{n}} to the absolutely convergent series
  
  ∑ (λ n − λ n + 1) B n. {\displaystyle \sum (\lambda _{n}-\lambda _{n+1})\,B_{n}.}