Alternating Series Convergence Test

[cp255]

According to my calculus book two parts to testing an alternating series for convergence. Let s = Ʃ(-1)ⁿ b_n. The first is that b_{n + 1} < b_n. The second is that the lim_{n$\rightarrow$∞} b_n = 0. However, isn't the first condition unnecessary since b_n must be decreasing if the limit is zero. I can't think of any example where the limit will be zero and the function is increasing (assuming b_n is not negative which would not really make sense since the series is alternating).

 

[micromass]

What if $b_{2n} = 0$ and $b_{2n+1} = \frac{1}{2n+1}$. That will give a divergent series.

 

[WannabeNewton]

$$\frac{sin^{2}(n)}{n}$$ is not monotone decreasing.

 

[cp255]

Yes but although sin²(x)/x is not always decreasing the series still converges. I just don't really see the point of the first test. Especially since this series seems to converge and is at times increasing.

 

[micromass]

Yes but although sin²(x)/x is not always decreasing the series still converges. I just don't really see the point of the first test. Especially since this series seems to converge and is at times increasing.

A test will never work for all series. For any convergence test, there will be a convergent series that is not described by the test.

The alternating series test is handy because it says, for example, that

$$\sum \frac{(-1)^n}{n}$$

converges. Furthermore, it gives a handy approximation that if $\sum (-1)^n b_n$ is your series, then it converges to x and

$$\left|x - \sum_{n=1}^k (-1) b_n\right|\leq b_k$$

So it tells you how close you are to your limit.

 

[WannabeNewton]

Yes but although sin²(x)/x is not always decreasing the series still converges. I just don't really see the point of the first test. Especially since this series seems to converge and is at times increasing.

You asked for a sequence which is not strictly monotone decreasing but still satisfies $\lim_{n\rightarrow \infty}a_n = 0$. The above is such an example. Note that not strictly monotone decreasing doesn't imply strictly monotone increasing. A sequence can be neither monotone increasing nor decreasing.

However, isn't the first condition unnecessary since b_n must be decreasing if the limit is zero.

 

[cp255]

You asked for a sequence which is not strictly monotone decreasing but still satisfies limn→∞an=0. The above is such an example. Note that not strictly monotone decreasing doesn't imply strictly monotone increasing. A sequence can be neither monotone increasing nor decreasing.

I understand this, but what I am wondering is why my book says that each successive term must be smaller than the last. Shouldn't all alternating series in which b_n goes to 0 as n approaches infinity be convergent? If this is true then what is the point of the first condition?

 

[micromass]

I understand this, but what I am wondering is why my book says that each successive term must be smaller than the last. Shouldn't all alternating series in which b_n goes to 0 as n approaches infinity be convergent? If this is true then what is the point of the first condition?

See post 2 for a counterexample.