limits - Sum of all the positive integers problem

The staff of Numberphile has shown that the sum of all the integers from $0$ to $\infty$ is $-\frac1{12}$. Recently I was looking for the sum of all the (positive) integers from $0$ to $n$ and I found that: $$\sum_{i=0}^n i=\frac{n(n+1)}{2}$$ So I decided to take the limit: $$\lim_{n\to \infty}\frac{n(n+1)}{2}$$ but that tends towards $\infty$ when I expected that to be $-\frac1{12}$!
Where did I got wrong? (the result is also confirmed by Wolfram Alpha)