alternating serries test

**superdude** · August 1st 2009, 05:33 PM

Is the convererse true for the alternating series test?
For example if $b_{n+1} \leq b_n$ is not true and $\lim_{n\to0} b_n=0$ is not true, does that mean the series diverges?

**skeeter** · August 1st 2009, 05:38 PM

Originally Posted by superdude

Is the convererse true for the alternating series test?
For example if $b_{n+1} \le b_n$ is not true and $\lim_{n\to \infty}b_n=0$ is not true, does that mean the series diverges?

edited the limit and latex errors ... am I correct?

any series whose nth term does not go to zero diverges.

**CaptainBlack** · August 1st 2009, 10:49 PM

Originally Posted by superdude

Is the convererse true for the alternating series test?
For example if $b_{n+1} \leq b_n$ is not true and $\lim_{n\to0} b_n=0$ is not true, does that mean the series diverges?

If $\lim_{n\to 0} b_n=0$ is not true then the sequence of partial sums cannot converge.

CB

**CaptainBlack** · August 1st 2009, 10:55 PM

Originally Posted by superdude

Is the convererse true for the alternating series test?
For example if $b_{n+1} \leq b_n$ is not true and $\lim_{n\to0} b_n=0$ is not true, does that mean the series diverges?

if $b_{n+1} \leq b_n$ is not true but $\lim_{n\to0} b_n=0$ is true then the series may converge.

Consider:

$S=\sum_{n=0}^{\infty} a_n$

$a_n = \begin{cases} (1/3)^n & n \text{ even} \\ -(1/2)^{n-1} & n \text{ odd} \end{cases}$

CB

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