I have read chapter about definite integrals from one textbook. When it came to definitiion of definite integral, there was this formula:
For all $\varepsilon > 0$ there exists $\delta > 0$ such that, for any tagged partition $\left [ a, b \right ]$ with mesh less than $\delta$, we have
$$\left | \sum_{i=1}^{n} f(t_i)\Delta_{t_{i}} -S \right |\leq \varepsilon.$$
I know that the sum from this formula is a Riemann sum.
Can someone explain what is the meaning of absolute value and $\varepsilon$ in this formula? Why are we subtracting $S$ from the Riemann sum?
We're subtracting $S$ from the sum (it doesn't really matter if we were instead subtracting the sum from $S,$ as you'll soon see) because we want to estimate the "difference" between them; in actual fact, we care about the absolute difference between them, without caring which is bigger or smaller. And why do we care about the distance between these two quantities? Because it tells us how close they are. In other words, it's a measure of how good an approximation to $S$ the Riemann sum is. First, note that distance cannot be negative. This is what the absolute value symbol indicates.
Now we can talk about what the statement means. Given an arbitrary positive tolerance for error $\epsilon,$ it says that we can approximate $S$ by the Riemann sum to within this tolerance by making the mesh sufficiently fine (specifically, to within $\delta >0.$)