section_path: Properties of series/Evaluation of truncation errors/Alternating series
section_name: Alternating series
paragraph_text: Alternating series
  
  When conditions of the alternating series test are satisfied by S:= ∑ m = 0 ∞ (− 1) m u m {\textstyle S:=\sum _{m=0}^{\infty }(-1)^{m}u_{m}}, there is an exact error evaluation. Set s n {\displaystyle s_{n}} to be the partial sum s n:= ∑ m = 0 n (− 1) m u m {\textstyle s_{n}:=\sum _{m=0}^{n}(-1)^{m}u_{m}} of the given alternating series S {\displaystyle S}. Then the next inequality holds: | S − s n | ≤ u n + 1. {\displaystyle |S-s_{n}|\leq u_{n+1}.}