The alternating series test (AST) for a sequence $\sum_{n=1}^{\infty} (-1)^n b_{n}$ stipulates that $\{b_{n}\}$ must be decreasing. Paul's Online Math Notes say that $\{b_{n}\}$ must be decreasing for large $n$, while most other sources (e.g. 1, 2, 3) say that $\{b_{n}\}$ must be decreasing for all $n$. Which of these is correct, and why? An example would help to clarify this.
Alternating Series Test: must b_n be decreasing for all n or only for large n
divergent-seriessequences-and-series
Best Answer
It doesn't matter since converegence of a series is determined only by the behaviour at the tails, i.e. large $n$. Any truncation of the series at a finite $N$ will be a finite number...