Convergence of $\sum_{n=1}^{\infty} n$ and integral test

Question

I have read online, that one can show that $\sum_{n=1}^{\infty}n = -\frac{1}{12}$.

But isn't this a Riemann Series of the form $\sum_{n=1}^{\infty} \frac{1}{n^p}$, where $p=-1$.

And if so, can't you prove that

$ \lim_{N \to \infty} \int^N \frac{1}{x^{-1}} = \lim_{N \to \infty} \frac{x^2}{2} = \infty$,

thus, by the integral test for convergence, the series diverges? Where is my mistake?

Answer 1

I think a requirement of the integral test is that $ f (n) $ be monotone decreasing.

Check http://www.math.com/tables/expansion/tests.htm. Under integral test.