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I would expect the following sum to converge:

$$ \sum_{n=-\infty}^\infty n=0 $$

as if I define its partial sums as a series:

$$ b_k= \sum_{n=-k}^k n $$

all of the terms in the series $b_k$ are equal to $0$ identically, and of course the series has a limit of $0$.

Also the integral $\int_{-\infty}^\infty xdx=0$ at least in the limit sense, though I'm not sure if the integral test is valid here.

Am I correct or does the sum diverge? My calculation relies on the fact that I took the upper and lower bound of the partial sums to infinity together. Is it valid, or must I allow them to approach infinity independently (which means my sum diverges)?

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Defining $\displaystyle\int_{-\infty}^{\infty}xdx=\lim_{M\rightarrow\infty}\displaystyle\int_{-M}^{M}xdx=0$ is called the principal value sense. Most of the cases, if it were not stated, we are not taking principal value sense, rather, to be $\displaystyle\int_{-\infty}^{\infty}xdx=\lim_{M,N\rightarrow\infty}\int_{-M}^{N}xdx$, which does not exist, this is called Improper Riemann integral.

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  • $\begingroup$ So if I understand you correctly, unless specify noted the standard definition is taking both bounds to infinity independently. Does this apply to the discrete series too? $\endgroup$
    – Yair M
    Nov 2, 2017 at 14:50
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    $\begingroup$ Yes, true. To have the symmetric one, one says, principal value sense as I have seen in most of the literature. $\endgroup$
    – user284331
    Nov 2, 2017 at 14:51

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