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:
- Positiveness: ( a_n > 0 ) for all ( n ).
- Decreasing: ( a_{n+1} \leq a_n ) for all ( n ) (the sequence ( a_n ) is decreasing).
- Range: ( \lim_{n \to \infty} a_n = 0 ).
If all three conditions are met, then the series (S) converges.