Alternating series - alternating series

Japanese: 交項級数 - こうこうきゅうすう（英語表記）alternating series

When a ₁ , a ₂ , ..., a _n , ... are all ≧0, a series of the form is called an alternating series. In other words, an alternating series is a series in which positive and negative terms appear alternately, in the form of a ₁ - a ₂ + a ₃ - a ₄ + ..., and is also called an alternating series. If the sequence { a _n } is monotonically decreasing, and if a _n →0 when n →∞, then converges. This is called Leibniz's theorem. For example, the series and diverge, but converge. It is known that the sums of these series are log2 and π/4, respectively.

Source: Heibonsha World Encyclopedia, 2nd Edition Information

Japanese:

a₁,a₂，……，a_n，……がすべて≧0のとき，の形の級数を交項級数という。すなわち，交項級数とはa₁－a₂＋a₃－a₄＋……の形に，正の項と負の項が交互に現れる級数であって，交代級数とも呼ばれる。数列｛a_n｝が単調減少で，n→∞のときa_n→0となるならば，は収束する。これをライプニッツの定理という。たとえば，級数，は発散するが，は収束する。これらの級数の和は，それぞれlog2,π/4となることが知られている。

出典　株式会社平凡社世界大百科事典第２版について　情報

<<: Mining and manufacturing

>>: Koukouki - Koukouki