Unformatted text preview: One application of the definite integral is to find the area under a curve. If you can find a function that produces the terms of the series when evaluated along the natural numbers, then you can take the definite integral of that function and compare the two areas. If the definite integral evaluated from one to infinity (which is an improper integral ) diverges, then the series also diverges. Notice that in the diagram the area of the series is greater than the area under the curve. Since the area under the curve is infinite, the series would have to be infinite too. However, if the definite integral evaluated from one to infinity converges, then the series converges. Since the area under the curve is finite, the value of the series from the second through the final term is also finite. Adding in the first term still leaves you with a finite value....
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 Spring '10
 hyon
 Calculus, Indian mathematics

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