## 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 limx→∞ 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 ∞.

Last Update: 2006-11-07