{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# PP 12.3 - The Integral Test and Estimates of Sums Comments...

This preview shows pages 1–7. Sign up to view the full content.

The Integral Test and Estimates of Sums

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Comments In general, finding the sum of an infinite series is not easy to do. Since knowing whether a series is convergent when it appears as part of the solution of an application problem is important, we need to develop tools to determine the behavior of a series. These tools are tests based on the behavior of improper integrals or series of known behavior.
THE INTEGRAL TEST 1 n n a = 1 ( ) f x dx Suppose f is a continuous, positive, decreasing function on [1, ) and let a n = f ( n ). Then, the series only if is convergent. is convergent if and In other words, the series and the improper integral have the same behavior.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Illustrating the Theorem Consider the series 2 2 2 2 2 2 1 1 1 1 1 1 1 ... 1 2 3 4 5 n n = = + + + + + There’s no simple formula for the sum s n of the first n terms, so finding if the series is convergent can NOT be done through the definition.
This figure shows the curve y = 1/ x 2 and rectangles that lie below the curve. The base of each rectangle is an interval of length 1. The height is equal to the value of the function y = 1/ x 2 at the right endpoint of the interval.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Thus, the sum of the areas of the rectangles is: 2 2 2 2 2 2 1 1 1 1 1 1 1 ...
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}