I have the following question from Function Theory of One Complex Variable – Greene/Krantz:
Give an example of a series of complex coefficients $ a_n$ such that $\lim_{N \to + \infty} \sum_{n= -N}^{N} a_n$ exists but $\sum_{-\infty}^{+\infty} a_n$ does not converge.
The answer key I have says that $a_n = n$ answers the question.
I understand that $\lim_{N \to + \infty} \sum_{n= -N}^{N} n = \lim_{N\to+\infty}[-N + (-N+1) +…+ -1 +0+1+…(N-1)+N] = 0$,
as can be seen from each term cancelling.
However, on the question of why $\sum_{-\infty}^{+\infty} n$ does not converge I'm a little stumped. Thinking about it intuitively, wouldn't you expect the same sort of cancellation of terms?
If anyone can provide a formal proof of why this series doesn't converge, I'd be very grateful. Thanks in advance!
Best Answer
This is similar to the difference between integral and Cauchy principal value
For example you know that $$\int _ {\mathbb R} x dx$$ does not exists, but
$$\lim_{R \to \infty} \int_{-R}^R x dx = 0$$
which is the Cauchy principal value.
The main point is that $\int_\mathbb R$ is not defined as a limit $\lim_{R \to \infty} \int_{-R}^R$.. For the riemann integral it is defined as
$$\lim_{a \to -\infty} \int_a^0 x dx + \lim_{b \to +\infty} \int_{0}^b x dx$$
and you can already see that neither of those converge.
A similar difference is there with the Lebesgue integral; The integral of a function $f(x)$ is defined as
$$\int f(x) dx = \int f(x) ^+ dx - \int f(x)^-dx$$ where $f(x)^+ = \max(f(x), 0)$ and $f(x)^- =- \min(f(x), 0)$. The main point is that we want the two separate integral to exists, and only then (if both exists) we sum them up. In this way there can't be any cancellation like the one happening with Cauchy principal value.
Your case is clearly in the same spirit, although it depends on how you define the double series $\sum_{-\infty}^{\infty}$. This is either defined as a double limit (in the "spirit" of the riemann integral) or as an integral with respect to the counting measure on $\mathbb Z$ (so basically it's a Lebesgue integral)