Conditional Convergence

In document Introduction to Real Analysis (Page 171-173)

We say thatf isnonoscillatoryatb .D 1 if b D 1/iff is defined onŒa; b/and does not change sign on some subintervalŒa1; b/ofŒa; b/. Iff changes sign on every such subinterval,f isoscillatoryatb . For a function that is locally integrable onŒa; b/ and nonoscillatory atb , convergence and absolute convergence ofRabf .x/ dx amount to the same thing (Exercise3.4.16), so absolute convergence is not an interesting concept in connection with such functions. However, an oscillatory function may be integrable, but not absolutely integrable, onŒa; b/, as the next example shows. We then say thatf is conditionallyintegrable onŒa; b/, and thatRabf .x/ dxconvergesconditionally.

Example 3.4.14

We saw in Example3.4.13that the integral

I.p/D

Z 1

1 sinx

xp dx

is not absolutely convergent if0 < p1. We will show that it converges conditionally for these values ofp.

Integration by parts yields Z c 1 sinx xp dxD cosc cp Ccos1 p Z c 1 cosx xpC1dx: (3.4.8) Since ˇ ˇ ˇcosx xpC1 ˇ ˇ ˇ 1 xpC1

andR11x p 1dx <1ifp > 0, Theorem3.4.6implies thatx p 1cosx is absolutely integrableŒ1;1/ifp > 0. Therefore, Theorem3.4.9implies thatx p 1cosxis integrable

Œ1;1/ifp > 0. Lettingc! 1in (3.4.8), we find thatI.p/converges, and

I.p/Dcos1 p

Z 1

1 cosx

xpC1dx if p > 0: This and (3.4.7) imply thatI.p/converges conditionally if0 < p1.

The method used in Example3.4.14is a special case of the following test for convergence of improper integrals.

Theorem 3.4.10 (Dirichlet’s Test)

Suppose thatf is continuous and its an- tiderivativeF .x/ D Raxf .t / dt is bounded onŒa; b/:Letg0 be absolutely integrable on

Œa; b/;and suppose that

lim

x!b g.x/D0: (3.4.9)

ThenRabf .x/g.x/ dxconverges:

Proof

The continuous functionfgis locally integrable onŒa; b/. Integration by parts yields Z c a f .x/g.x/ dxDF .c/g.c/ Z c a F .x/g0.x/ dx; ac < b: (3.4.10) Theorem3.4.6implies that the integral on the right converges absolutely asc!b , since Rb

a jg0.x/jdx <1by assumption, and

jF .x/g0.x/j Mjg0.x/j;

whereM is an upper bound forjFjonŒa; b/. Moreover, (3.4.9) and the boundedness ofF imply that limc!b F .c/g.c/D0. Lettingc!b in (3.4.10) yields

Z b a f .x/g.x/ dx D Z b a F .x/g0.x/ dx; where the integral on the right converges absolutely.

Dirichlet’s test is useful only iff is oscillatory atb , since it can be shown that iff is nonoscillatory atb andF is bounded onŒa; b/, thenRabjf .x/g.x/jdx <1if onlygis locally integrable and bounded onŒa; b/(Exercise3.4.14).

Example 3.4.15

Dirichlet’s test can also be used to show that certain integrals di-

verge. For example, Z

1

1

xqsinx dx

diverges ifq > 0, but none of the other tests that we have studied so far implies this. It is not enough to argue that the integrand does not approach zero asx ! 1(a common mistake), since this does not imply divergence (Exercise4.4.31). To see that the integral diverges, we observe that if it converged for someq > 0, thenF .x/ D R1xxqsinx dx would be bounded onŒ1;1/, and we could let

f .x/Dxqsinx and g.x/Dx q in Theorem3.4.10and conclude that

Z 1

1

sinx dx

The method used in Example3.4.15is a special case of the following test for divergence of improper integrals.

Theorem 3.4.11

Suppose thatuis continuous onŒa; b/andRabu.x/ dxdiverges:Let

vbe positive and differentiable onŒa; b/;and suppose thatlimx!b v.x/D 1andv0=v2 is absolutely integrable onŒa; b/:ThenRabu.x/v.x/ dxdiverges:

Proof

The proof is by contradiction. Letf D uv andg D 1=v, and suppose that Rb

a u.x/v.x/ dxconverges. Thenf has the bounded antiderivativeF .x/D

Rx

a u.t /v.t / dt onŒa; b/, limx!1g.x/D0andg0D v0=v2is absolutely integrable onŒa; b/. Therefore,

Theorem3.4.10implies thatRabu.x/ dxconverges, a contradiction.

If Dirichlet’s test shows thatRabf .x/g.x/ dxconverges, there remains the question of whether it converges absolutely or conditionally. The next theorem sometimes answers this question. Its proof can be modeled after the method of Example3.4.13(Exercise3.4.17). The idea of an infinite sequence, which we will discuss in Section 4.1, enters into the statement of this theorem. We assume that you recall the concept sufficiently well from calculus to understand the meaning of the theorem.

Theorem 3.4.12

Suppose thatgis monotonic onŒa; b/andRabg.x/ dx D 1:Letf

be locally integrable onŒa; b/and

Z xjC1 xj

jf .x/jdx; j 0;

for some positive;wherefxjgis an increasing infinite sequence of points inŒa; b/such thatlimj!1xj DbandxjC1 xj M; j 0;for someM:Then

Z b a j

f .x/g.x/jdx D 1:

In document Introduction to Real Analysis (Page 171-173)