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Prove directly from the definition of series convergence that $$\sum_{n=1}^\infty (-1)^n$$ diverges.

In this exam question it has not been stated which definition to use, however I am assuming the ratio test could be used to prove divergence?

If this is correct how would I go about doing this?

Thanks in advance!

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How will you grant us the 8 marks? –  Hagen von Eitzen Aug 20 '13 at 20:29
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Compute the first few partial sums, and you will see what happens. –  mrf Aug 20 '13 at 20:31
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@HagenvonEitzen one upvote and one downvote would make 8 marks! –  mrf Aug 20 '13 at 20:32
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To use the ratio test you would have to look at $\lim_{\to \infty}|{a_{n+1} \over a_n}|$. Since this limit is $1$ for your series, the ratio test is inconclusive. –  walcher Aug 20 '13 at 20:33
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Until you see the pattern. –  mrf Aug 20 '13 at 20:35
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3 Answers

up vote 4 down vote accepted

By definition, $\sum_{n=1}^\infty a_N$ converges iff there exists $a$ such that $\lim_{N\to\infty}\sum_{n=1}^N a_n = a$, that is for every $\epsilon>0$ there exists ...

And the series diverges if it does not converge.

You are to use this instead of any (derived) tests.

Hint for the solution:

With $\epsilon=\frac12$, assume that $\left|\sum_{n=1}^N a_n-a\right|<\epsilon$. What can you say about $\left|\sum_{n=1}^{N+1} a_n-a\right|<\epsilon$?

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Hint: Construct the sequence of partial sums and show that it doesn't converge.

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njguliyev and Hagen von Eitzen have shown the correct approach. I just want to belabor a point, in case it was too subtle. :) By definition, a series converges if its sequence of partial sums converges. That is THE definition of series convergence.

All of the tests you've studied -- the term test, the ratio test, the root test, the condensation test, the alternating series test, even the Cauchy criterion -- are consequences of this definition. So if an exam says "Prove directly from the definition of series convergence", it is asking you NOT to apply any of the tests!

The purpose of the question is not to see whether you can solve the mathematical problem to determine whether a particular series converges. The purpose is to see whether you know the definition and whether you can apply it, without calling upon the most powerful tools you've been given. If you use any "test", you won't get full credit.

(As an aside, the ratio test happens to be inconclusive for this series anyway!)

Unfortunately, there's still a bit of a ambiguity. The sequence of partial sums is $(-1,0,-1,0,\ldots)$. Clearly it doesn't converge, but are you required to prove that as well? If so, are you required to argue directly from the epsilon-delta definition of a limit? Or are you allowed to call upon the theorems about sequences that you've studied? For example, is it good enough to observe that the sequence has two subsequences that converge to different limits? Only the grader for the exam knows for sure.

During a real exam, you could write down "This sequence doesn't converge" and then move on to the next question. When you finish the exam, if you still have time remaining, you can go back and provide a proof, just to be safe.

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