# Question about convergence of series

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)?

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.