Riemann sum
A Riemann sum is the staircase approximation to the area under a graph: pile up a rectangle over each strip, each as tall as the function at one sampled point inside that strip, and add the rectangle areas. If the strips are thin enough, the jagged staircase hugs the true region, and its total area approaches the integral.
Given a tagged partition (P, T) of [a, b] with P = {x_0, ..., x_n} and tags t_k in [x_(k-1), x_k], the corresponding Riemann sum of f is S(f, P, T) = sum_{k=1}^{n} f(t_k)(x_k - x_(k-1)). Each term f(t_k) times the width is the signed area of one rectangle, signed because f(t_k) may be negative.
We say the Riemann sums converge to a number I if for every epsilon > 0 there is a delta > 0 such that every tagged partition with mesh < delta gives |S(f, P, T) - I| < epsilon, no matter how the tags are chosen. When such an I exists it is the Riemann integral of f. This is a genuine limit over a directed family, not a sequence limit, because the tags are quantified universally.
For f(x) = x^2 on [0, 1] with the uniform partition and right endpoints, the sum is sum_{k=1}^{n} (k/n)^2 (1/n) = (n+1)(2n+1)/(6n^2), which tends to 1/3 as n -> infinity, matching the integral of x^2 from 0 to 1.
A closed-form Riemann sum whose limit reproduces the familiar integral value 1/3.
Left-endpoint, right-endpoint, and midpoint rules are all Riemann sums for particular tag choices; they are the seeds of basic numerical integration. Convergence of one particular sequence of sums does not by itself prove integrability — every tag choice must be controlled simultaneously.