This is a conjecture I made while studying the summation of series, and it has now been confirmed as correct.
Suppose there exists a series \sum_{x=1}^{\infty} f(x). If
\lim_{x \to \infty} \int f(x)dx \to \infty
then the series diverges.
If \lim_{x \to \infty} \int f(x)dx converges, then the series converges.
Examples:
For the series \sum_{x=1}^{\infty} 1/x, since \int (1/x)dx = \ln x and \lim_{x \to \infty} \ln x \to \infty, the series diverges.
For the series \sum_{x=1}^{\infty} 1/{x^2}, since \int (1/{x^2})dx = -1/x and \lim_{x \to \infty} -1/x \to 0, the series converges.
It turns out this conclusion already exists; I am now presenting it in a definitive form.
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