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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 $n>3$.)

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Eventually decreasing is sufficient. This follows from the fact that convergence of a series is not affected by its first few terms. –  David Mitra May 14 '12 at 13:30
    
Think about Riemann criterion. –  Monoide May 14 '12 at 13:36
    
@Monoide Why should they? –  Did Jan 14 '13 at 17:39
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1 Answer

up vote 4 down vote accepted

Are you considering the series $\sum\limits_{n=1}^\infty (-1)^n{\sqrt n\over n+4}$?

In the criteria for the Alternating Series Test, the positive terms being eventually decreasing to 0 is sufficient for convergence of the series. This follows from the fact that convergence of a series is not affected by its first few terms.

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.

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Oh yes, of course-- the convergence of a series is not affected by altering a finite number of its terms. Thank you for clearing that up! –  Ryan May 14 '12 at 14:00
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