Question about Alternating Series Test

[DryRun]

Homework Statement
After reading a few topics on this forum, i just realized that i had misunderstood the sequence v/s series theorem when it concerns the alternating sign. So, i went back to my notes, and I'm surprised to see that there is no mention of a series test. There is only a sequence test for alternating sign where i need to take the modulus. I guess my tutor skipped that part, or i lost my sheets. Either way, I'm now trying to learn the concept by myself.

The attempt at a solution

Here is what I've understood:
The AST is used to test for a series' convergence.

The first step is to check if: \lim_{n\rightarrow \infty} a_n = 0
The second check is if: a_{n+1} \leq a_n
I got those equations online (youtube), so i hope these are correct. However, i have several doubts:

What if the first check is OK but the second check is not. Does that mean that the series diverges?

In the example that I've seen in some video, i see that if the first check fails, then it automatically diverges. But I'm wondering if that's always the case?

 

[Ray Vickson]

Homework Statement
After reading a few topics on this forum, i just realized that i had misunderstood the sequence v/s series theorem when it concerns the alternating sign. So, i went back to my notes, and I'm surprised to see that there is no mention of a series test. There is only a sequence test for alternating sign where i need to take the modulus. I guess my tutor skipped that part, or i lost my sheets. Either way, I'm now trying to learn the concept by myself.

The attempt at a solution

Here is what I've understood:
The AST is used to test for a series' convergence.

The first step is to check if: \lim_{n\rightarrow \infty} a_n = 0
The second check is if: a_{n+1} \leq a_n
I got those equations online (youtube), so i hope these are correct. However, i have several doubts:

What if the first check is OK but the second check is not. Does that mean that the series diverges?

In the example that I've seen in some video, i see that if the first check fails, then it automatically diverges. But I'm wondering if that's always the case?

It does not matter if a_{n+1} \leq a_n for all n, but only that it happens for all sufficiently large n (that is, it might fail for the first few terms, but must happen from all n beyond a certain finite value).

As to your second question: the condition you cite is a sufficient condition for convergence, not a necessary one. For example, the series \sum (-1)^n t_n,, where t_n = \left\{ \begin{array}{cl}1/n^2,&amp; n \text{ even} \\<br /> 2/n^2 ,&amp; n \text{ odd} \end{array} \right.
is convergent, but does not satisfy the inequality.

RGV

 

[DryRun]

I'm not sure... So, the AST is synonymous to the nth-term test for divergence, in that, the latter can only be used for testing divergence but says nothing about the series' convergence. Similarly, the AST is only for testing convergence, and it cannot account for divergence. Correct?

It does not matter if a_{n+1} \leq a_n for all n, but only that it happens for all sufficiently large n (that is, it might fail for the first few terms, but must happen from all n beyond a certain finite value).

As to your second question: the condition you cite is a sufficient condition for convergence, not a necessary one. For example, the series \sum (-1)^n t_n,, where t_n = \left\{ \begin{array}{cl}1/n^2,&amp; n \text{ even} \\<br /> 2/n^2 ,&amp; n \text{ odd} \end{array} \right.
is convergent, but does not satisfy the inequality.

RGV

In your example, i don't understand why n is odd or even? What's your point?

 

[Mark44]

I wouldn't say that the alternating series test is synonymous to the n-th term test, but they are similar in a way.

The alternating series test says that under certain conditions, an alternating series converges. It does not mention divergence.

The nth term test says that under certain conditions, a series diverges. It does not mention convergence.

Ray's example with different terms for odd or even n shows a series for which lim t_n = 0, but the series is not monotonically decreasing.

Edit:
An important point about these theorems is the idea of what an implication means in mathematics. Many calculus students are not used to thinking logically, so the fine points about mathematical logic go right over their heads.

Let's look at an example. Suppose I tell you, "If you mow my lawn this Saturday, then I'll give you $100."

As long as the condition (you mow the lawn) is met, you are justified in expecting to be paid. If the condition is not met, you cannot reasonably conclude that I will pay you, and you also cannot conclude that I won't pay you. (Out of the kindness of my heart, I might just give you $100.)

My point is - make sure that the hypotheses (conditions) of a theorem are met, then the conclusion inescapably must follow. If the hypetheses are not met, anything can happen.

Case in point: nth term test, with series Ʃ (1/n). lim a_n = 0. The hypothesis of this theorem is not met, so the theorem does not apply.[/color]

 

[DryRun]

I think i finally understand the conditions for which AST applies. The 'lawn mowing' example is a very practical way of untangling the ideas associated with the theorem.

Thank you very much for your help, Mark44 and RGV.