Alternating Series Test/Test for Divergence

[Auskur]

So I've been practicing several series that can be solved using the alternating series test, but I've came to a question that's been bothering me for sometime now.

If a series fails the alternating series test, will the test for divergence always prove it to be divergent?

Typically, in most examples that I find on the James Stewart website they completely omit the alternating component when using the test for divergence. Is this because it will make the limit not exist?

Some examples.

Ʃ n = 1 to ∞ {(-1)^n * 2^(1/n)}

The alternating series test fails, so in the solution they take the
lim n → ∞ 2^(1/n) = 1.

Ʃ n = 1 to ∞ {(-1)^n * ((2^n) / n^2))}

Similarly, the alternating series test fails, so they use the test for divergence.
Again, they fail to include the (-1)^n and conclude that

lim n → ∞ 2^n / n^2 = ∞ ∴ the series is divergent.

Why do they not include the (-1)^n, won't this make the limit not exist? Obviously, this will still prove the series to diverge, but which one is the correct way to do it? Should I write the limit does not exist or the limit = 1? Thanks in advance to anyone that can help me out!

 

[micromass]

You are right. They need to include the $(-1)^n$ in the limit. So you need to find the limit

$$\lim_{n\rightarrow +\infty} (-1)^n 2^{1/n}$$

which indeed won't exist.

However, it is obvious that $\lim_{n\rightarrow +\infty} x_n=0$ if and only if $\lim_{n\rightarrow +\infty} |x_n|=0$. This shows that $$\lim_{n\rightarrow +\infty} (-1)^n 2^{1/n}$$ is nonzero because $\lim_{n\rightarrow +\infty} 2^{1/n}$ is nonzero.

So, they are also right. Instead of calculating the limit of the terms of the series, they calculate the limit of the absolute value. And because of that absolute value, the $(-1)^n$ factor disappears.
I do feel that they should mention this explicitely.

 

[Auskur]

Thank you. That was helpful : )

 

[Dartx4]

Just realize that an alternating series is only alternating when it has (-1)^n or (-1)^n+1. So if they posses either of these then you have to figure out what the entire summation is without that part. You then call it (a-sub n) the lim n→∞ of that function. It then has to equal 0 and be greater than (a-sub n+1). If it follows both of these then it will converge, if it doesn't then it diverges.

With this being said. Your example shows an alternating series, but when you take the lim n→+∞ you get 1. This is not 0 so you immediatly say the series is diverging.

 

[HallsofIvy]

Notice, by the way, that to prove a series, $\sum a_n$ divergent by the "divergence test" it is sufficient to prove that $lim_{n\to\infty} a_n$ is not 0. In fact, if an alternating series does not converge, then that limit does not even exist!