# How can I find out if a non-convergent series is “indeterminate” (that is, “oscillating”) or “divergent”?

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?

• If $a_n\to l\neq 0$ then it is divergent; $s_n$ either converges to $+\infty$ or $-\infty$, depending on the sign of $l$. – Thomas Andrews May 20 '15 at 16:42
• If $a_n\to \ell\neq 0$, then all but finitely many $a_n$ are of the same sign as $\ell$. – Thomas Andrews May 20 '15 at 16:44
• @MichaelHardy That's why I've added all the definitions. – math-fun May 20 '15 at 16:52
• 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. – Thomas Andrews May 20 '15 at 16:55
• @ThomasAndrews Actually, I've made an effort before asking the question and I'm still making an effort right now. – math-fun May 20 '15 at 17:06

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 $+\infty$.
• 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? – math-fun May 20 '15 at 22:42