# Alternating Series Test for Convergence - can the sequence be initially non-decreasing?

I was given the following question:

Determine if the following series is convergent. You may use basic series, but you should clearly state which results or rules you use.

$$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}\sqrt{n}}{n+4}$$

Does the Alternating Series Test require the positive term $\frac{\sqrt n}{n+4}$ to be decreasing for all $n$ or is it sufficient that the term is eventually decreasing? ($\frac{\sqrt n}{n+4}$ is decreasing only for $n>3$.)

• Eventually decreasing is sufficient. This follows from the fact that convergence of a series is not affected by its first few terms. Commented May 14, 2012 at 13:30
• Think about Riemann criterion. Commented May 14, 2012 at 13:36
• @Monoide Why should they?
– Did
Commented Jan 14, 2013 at 17:39

Are you considering the series $\sum\limits_{n=1}^\infty (-1)^n{\sqrt n\over n+4}$?
So, you could argue that $\sum\limits_{n=1}^\infty (-1)^n{\sqrt n\over n+4}$ converges if and only if $\sum\limits_{n=4}^\infty (-1)^n{\sqrt n\over n+4}$ converges. The latter series converges by the Alternating Series Test; thus the former series converges.