You can see that since the degrees of the denominators is greater than that of the numerators, they dominate (for each term).
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?