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Definitions:

Given a sequence $\{a_n\}$, define $$s_n= \sum_{j=0}^n a_j.$$ The sequence $\{s_n\}$ is called the series of partial sums of $\{a_n\}$. A series is convergent if $\{s_n\}$ has finite limit; devergent if $\{s_n\}$ has an infinite limit; and indeterminate if $\{s_n\}$ has no limit.

Known facts:

We have some criteria to determine convergence or divergence of a series if $\{s_n\}$ are all positive (or all negative) for $n$ greater than a certain index $n_0$ (ratio, root, integral tests, etc) or if $\{s_n\}$ has alterning signs (Liebnitz test).

Problems and questions:

Suppose that a series $ \sum a$ has not all non-negative (or non-positive) terms. Suppose also that we know (for example, by seeing that $a_n \to l \neq 0$) that it is not-convergent; what are some strategies to deduce if the series is divergent or indeterminate?

In particular, is it true that if $a_n \to l \in \mathbb{R} \cup \{\pm \infty\}$ then the series diverge and if $\lim_n a_n$ doesn't exist then the series is indeterminate? Why?

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    $\begingroup$ If $a_n\to l\neq 0$ then it is divergent; $s_n$ either converges to $+\infty$ or $-\infty$, depending on the sign of $l$. $\endgroup$ – Thomas Andrews May 20 '15 at 16:42
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    $\begingroup$ If $a_n\to \ell\neq 0$, then all but finitely many $a_n$ are of the same sign as $\ell$. $\endgroup$ – Thomas Andrews May 20 '15 at 16:44
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    $\begingroup$ @MichaelHardy That's why I've added all the definitions. $\endgroup$ – math-fun May 20 '15 at 16:52
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    $\begingroup$ Bored now. On to another question. Helping people is fun if they are making an effort, @math-fun, not when they are begging for you to give more. $\endgroup$ – Thomas Andrews May 20 '15 at 16:55
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    $\begingroup$ @ThomasAndrews Actually, I've made an effort before asking the question and I'm still making an effort right now. $\endgroup$ – math-fun May 20 '15 at 17:06
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The question is somewhat broad, but here's a partial answer: If the terms don't approach $0$ and they are all non-negative, then the series diverges to $+\infty$.

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  • $\begingroup$ I knew that: as I have specified in the question ("suppose a series does not belong to these groups"), I'm interested in the case when the series doesn't have all non-negative (or all non-positive) terms. Thank you anyway :). $\endgroup$ – math-fun May 20 '15 at 17:01
  • $\begingroup$ Is it true that if $a_n \to l \in \mathbb{R} \cup \{\pm \infty\}$ then the series $\sum a_n$ diverges and if $\lim_n a_n$ doesn't exist then the series is indeterminate? $\endgroup$ – math-fun May 20 '15 at 22:42

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