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## Main Question or Discussion Point

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, wont 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!

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, wont 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!

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