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Can I use the regular test for the convergence of series (for example the ratio test) to test if series converges when they start at $-\infty$ and run to $+\infty$ so over all the integers:

$$\sum_{\text{n}=-\infty}^\infty\text{a}_\text{n}=\sum_{\text{n}\in\mathbb{Z}}\text{a}_\text{n}$$

For the ratio test, we would have:

$$\lim_{\text{n}\to\infty}\left|\frac{\text{a}_{\text{n}+1}}{\text{a}_\text{n}}\right|$$

If I can't which kind of tests can I use indeed?

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    No, for instance it wouldn't work for $\sum_{n \in \Bbb Z} \frac1{2^n}$ – 2017-01-07
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    @OpenBall Yes I see, it give me $\frac{1}{2}$ when I compute the limit. – 2017-01-07
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    One splits the series and looks at $\sum\limits_{n = 0}^{\infty} a_n$ and $\sum\limits_{k = 1}^{\infty} a_{-k}$ separately. – 2017-01-07
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    Note that such a sumation is well defined surely for series of positive terms, but it need not to be well defined in general. Before asking how to show convergence, you should ask how to define convergence of such series. – 2017-01-07
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    @DanielFischer Can I split the series into:? $$\sum_{\text{n}\in\mathbb{Z}}\text{a}_\text{n}=\sum_{\text{n}=-\infty}^0\text{a}_\text{n}+\sum_{\text{n}=0}^\infty\text{a}_\text{n}$$ – 2017-01-07
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    That uses $a_0$ twice. Not a problem when you're just checking convergence, but you get a different result if $a_0 \neq 0$. You can, if you have a definition for $\sum\limits_{n = -\infty}^0 a_n$ in scope. Conventionally, that would be $\sum\limits_{k = 0}^{\infty} a_{-k}$ in the sense that we define convergence of the former as convergence of the latter, and the value in case of convergence ditto. All that is of course superfluous if you restrict your attention to summable families. – 2017-01-07
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    @DanielFischer Oh yes I see, so the second series should start at $\text{n}=1$ when $\text{a}_0\ne0$. But can I test the series apart from each other? – 2017-01-07
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    Depends on what sort of convergence you want to test for. Generally, you can split. You have summability if and only if you have absolute convergence, and you have $\sum\limits_{n\in\mathbb{Z}}\lvert a_n\rvert < +\infty$ if and only if both halves are absoutely convergent. For non-absolute convergence, the usual definition of convergence is via the split (the series converges by definition if and only if both halves converge). But one can also look at symmetric convergence - the series converges if and only if $p_N = \sum\limits_{n = -N}^N a_n$ converges (possibly with an additional – 2017-01-07
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    constraint that $a_n \to 0$ for $n \to \pm\infty$). This form of convergence is commonly considered for Fourier series. When you use symmetric convergence, you can't split. But then you are in the classical $-a_0 +\sum\limits_{n = 0}^{\infty} (a_n + a_{-n})$ setting anyway. – 2017-01-07

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