Elementary Calculus: Theorem 2: Integral Test.

The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages.

Home Infinite Series Series With Positive Terms Theorem 2: Integral Test.


Search the VIAS Library \| Index
Theorem 2: Integral Test. For our last test we need another theorem which is similar to Theorem 1. THEOREM 2 If the function F(x) increases for x ≥ 1, then lim_x→∞ F(x) either exists or is infinite. This says that the curve y - F(x) is either asymptotic to some horizontal line y = L or increases indefinitely, as illustrated in Figure 9.4.2. Figure 9.4.2 INTEGRAL TEST Suppose f is a continuous decreasing function and f(x) > 0 for all x ≥ 1. Then the improper integral and the infinite series either both converge or both diverge to ∞. Discussion Figure 9.4.3 suggests that so the series and the integral should both converge or both diverge to ∞. The Integral Test shows that the integral and the series have the same convergence properties. However, their values, when finite, are different. In fact, we can see from Figure 9.4.3(c) that the integral is less than the series sum, (a)(b)(c) Figure 9.4.3 The Integral Test PROOF As we can see from Figure 9.4.3, for each m we have The improper integral is defined by Since f(x) is always positive, the function dx is increasing, so by Theorem 2, the limit either exists or is infinite. Hence the improper integral either converges or diverges to ∞. Case 1 converges. For infinite H we have thus the infinite partial sum is finite. Hence the tail and the series converge. Case 2 diverges to ∞. Since , the infinite partial sum has infinite value, whence the series diverges to ∞.
Home Infinite Series Series With Positive Terms Theorem 2: Integral Test.

Last Update: 2006-11-07