section_path: History of the theory of infinite series/Semi-convergence
section_name: Semi-convergence
paragraph_text: Semi-convergence
  
  A series is said to be semi-convergent (or conditionally convergent) if it is convergent but not absolutely convergent.
  
  Semi-convergent series were studied by Poisson (1823), who also gave a general form for the remainder of the Maclaurin formula. The most important solution of the problem is due, however, to Jacobi (1834), who attacked the question of the remainder from a different standpoint and reached a different formula. This expression was also worked out, and another one given, by Malmsten (1847). Schlömilch (Zeitschrift, Vol.I, p. 192, 1856) also improved Jacobi's remainder, and showed the relation between the remainder and Bernoulli's function
  
  F (x) = 1 n + 2 n + ⋯ + (x − 1) n. {\displaystyle F(x)=1^{n}+2^{n}+\cdots +(x-1)^{n}.}
  
  Genocchi (1852) has further contributed to the theory.
  
  Among the early writers was Wronski, whose "loi suprême" (1815) was hardly recognized until Cayley (1873) brought it intoprominence.