You can see that since the degrees of the denominators is greater than that of the numerators, they dominate (for each term).
No, the sum is not equal to the integral.
Try this: Draw x and y coordinate axes. Sketch rectangles of height for n = 1, 2, 3, 4, 5 (say), where the base of each rectangle is the line segment from (n, 0) to (n+1, 0). Now sketch the curves and . is the total area of the first n rectangles. Can you "see" the inequality now?