Basic formulas and conditions associated with alternating series and the Leibniz criterion:

Alternating series

The alternating series has the form:
[ S = a_1 – a_2 + a_3 – a_4 + a_5 – \ldots ]
where ( a_n ) are the positive terms of the series.

Leibniz’s sign

For the convergence of an alternating series ( S = \sum_{n=1}^{\infty} (-1)^{n-1} a_n ) the following conditions must be met:

  1. Positiveness: ( a_n > 0 ) for all ( n ).
  2. Decreasing: ( a_{n+1} \leq a_n ) for all ( n ) (the sequence ( a_n ) is decreasing).
  3. Range: ( \lim_{n \to \infty} a_n = 0 ).

If all three conditions are met, then the series (S) converges.

От Math

Добавить комментарий

Ваш адрес email не будет опубликован. Обязательные поля помечены *