No Relation Between Absolute and Uniform Convergence We finally show the surprising fact that there are series that converge absolutely but not

No Relation Between Absolute and Uniform Convergence We finally show the surprising fact that there are series that converge absolutely but not

uniformly, and others that converge uniformly but not absolutely, so that there is no relation between the two concepts.

EXAMPLE 5 No Relation Between Absolute and Uniform Convergence

The series in Example 2 converges absolutely but not uniformly, as we have shown. On the other hand, the series

(x real)

converges uniformly on the whole real line but not absolutely. Proof. By the familiar Leibniz test of calculus (see App. A3.3) the remainder R n does not exceed its first term in absolute value, since we have a series of alternating terms whose absolute values form a monotone decreasing sequence with limit zero. Hence given P ⬎ 0, for all x we have

This proves uniform convergence, since N (P) does not depend on x.

The convergence is not absolute because for any fixed x we have

(⫺1) mⴚ1

where k is a suitable constant, and kS 1 >m diverges.

PROBLEM SET 15.5

(b) Power series. Study the nonuniformity of con- Fig. 368. Produce this exciting figure using your CAS.

1. CAS EXPERIMENT. Graphs of Partial Sums. (a)

vergence experimentally by graphing partial sums near Add further curves, say, those of s 256 ,s 1024 , etc. on the

the endpoints of the convergence interval for real same screen.

z⫽x .

SEC. 15.5 Uniform Convergence. Optional

(b) Termwise differentiation. Derive Theorem 4 Where does the power series converge uniformly? Give

POWER SERIES

from Theorem 3.

reason. (c) Subregions. Prove that uniform convergence of a

n⫹ 2 n series in a region G implies uniform convergence in 2. a a n

7n ⫺ 3 b z any portion of G. Is the converse true?

n⫽0 ⴥ

(d) Example 2. Find the precise region of convergence 1

3. (z ⫹ i) 2 n of the series in Example 2 with x replaced by a complex n⫽0 a 3 n

variable z.

2 3 ⬁ n (e) Figure 369. Show that x S m⫽1 (1 ⫹ x 2 (1 ⫺ i) ) ⴚm ⫽ 1 4. a (z ⫺ i) n

if x⫽ and 0 if

0 x⫽ 0

. Verify by computation that the

n⫽0

partial sums s 1 ,s 2 ,s 3 look as shown in Fig. 369. n 5. a n a 2 b (4z ⫹ 2i)

n⫽2

6. 1 a 2 (tanh n 2 ) z

Fig. 369. Sum s and partial

sums in Team Project 18(e)

n⫽1 a 2 (z ⫺ 2i)

HEAT EQUATION 10–17

UNIFORM CONVERGENCE

Show that (9) in Sec. 12.6 with coefficients (10) is a solution Prove that the series converges uniformly in the indicated

of the heat equation for t⬎ 0, assuming that f (x) is region.

continuous on the interval 0⬉x⬉L and has one-sided

z 2 n 10. 20 a derivatives at all interior points of that interval. Proceed as 2n! , ƒ z ƒ ⬉ 10 n⫽0 follows.

z n 19. Show that ƒB n ƒ is bounded, say ƒB n ƒ⬍K for all n. 11. a 2 , ƒzƒ⬉1 Conclude that

n⫽1 n

if t⭌t 0 ⬎ 0 12. , ƒzƒ⬉1

ƒu ƒ ⬍ Ke ⴚl n t 0 n

n 3 n⫽1 a cosh nƒzƒ and, by the Weierstrass test, the series (9) converges

sin n ƒzƒ uniformly with respect to x and t for t⭌t 0 , 0 ⬉ x ⬉ L. 13. a 2 , all z Using Theorem 2, show that u (x, t) is continuous for

n⫽1

n t⭌t 0 and thus satisfies the boundary conditions (2)

20. Show that ƒ 0u

(n!) 2 series of the expressions on the right converges, by 15. a z n , ƒzƒ⬉3

the ratio test. Conclude from this, the Weierstrass n⫽0 (2n!)

test, and Theorem 4 that the series (9) can be

tanh n ƒzƒ differentiated term by term with respect to t and the 16. a 0u

n (n ⫹ 1) , all z

resulting series has the sum >0t . Show that (9) can

n⫽1

be differentiated twice with respect to x and the

p n resulting series has the sum 0 2 2 2 n u >0x . Conclude from 17. a z , ƒ z ƒ ⬉ 0.56

this and the result to Prob. 19 that (9) is a solution n⫽1 n 4 of the heat equation for all t⭌t 0 . (The proof that (9)

satisfies the given initial condition can be found in (a) Weierstrass M-test. Give a proof.

18. TEAM PROJECT. Uniform Convergence.

Ref. [C10] listed in App. 1.)

CHAP. 15 Power Series, Taylor Series

CHAPTER 15 REVIEW QUESTIONS AND PROBLEMS

1. What is convergence test for series? State two tests from

(⫺2) n⫹1

memory. Give examples.

n⫽1 a 2. What is a power series? Why are these series very 2n important in complex analysis?

15. z n

RADIUS OF CONVERGENCE

3. What is absolute convergence? Conditional convergence? Uniform convergence?

Find the radius of convergence. Try to identify the sum of the series as a familiar function.

4. What do you know about convergence of power series?

z 5. What is a Taylor series? Give some basic examples. n

n⫽1 a n⫽0 a n ! series?

17. z n 6. What do you know about adding and multiplying power

16. n

(⫺1) n

7. Does every function have a Taylor series development? 18. a ( p z ) 2 n⫹1

n⫽0 (2n ⫹ 1)!

Explain.

z 19. a 20. a Maclaurin series? Give examples.

8. Can properties of functions be discovered from

n⫽0 (3 ⫹ 4i) 9. What do you know about termwise integration of series?

n⫽0 (2n)!

MACLAURIN SERIES

10. How did we obtain Taylor’s formula from Cauchy’s Find the Maclaurin series and its radius of convergence. formula?

Show details.

21. (sinh z 2 ) 2 >z

RADIUS OF CONVERGENCE

24. 1 Find the radius of convergence.

>(⫺z ) ⫺ 1) >z

n⫹ 1 11. a 2 (z ⫹ 1) n

n⫽2 n ⫹ 1 26–30

TAYLOR SERIES

4 n Find the Taylor series with the given point as center and its

12. a (z ⫺ p i ) n

n⫽2 n⫺ 1 radius of convergence.

SUMMARY OF CHAPTER 15

Power Series, Taylor Series

Sequences, series, and convergence tests are discussed in Sec. 15.1. A power series is of the form (Sec. 15.2)

z 0 is its center. The series (1) converges for ƒz⫺z 0 ƒ⬍R and diverges for ƒz⫺z 0 ƒ ⬎ R, where R is the radius of convergence. Some power series converge

Summary of Chapter 15

707

for all z (then we write R⫽⬁ ). In exceptional cases a power series may converge only at the center; such a series is practically useless. Also, R⫽ lim ƒ a n >a n⫹ 1 ƒ if this limit exists. The series (1) converges absolutely (Sec. 15.2) and uniformly (Sec. 15.5) in every closed disk ƒz⫺z 0 ƒ ⬉ r ⬍ R (R ⬎ 0). It represents an analytic

function f (z) for ƒz⫺z 0 ƒ ⬍ R. The derivatives f r (z), f s (z), Á are obtained by

termwise differentiation of (1), and these series have the same radius of convergence R as (1). See Sec. 15.3.

Conversely, every analytic function f (z) can be represented by power series. These

Taylor series of f (z) are of the form (Sec. 15.4)

(ƒz⫺z 0 ) ƒ ⬍ R),

n⫽0 n !

as in calculus. They converge for all z in the open disk with center z 0 and radius generally equal to the distance from z 0 to the nearest singularity of (point f (z) at which f (z) ceases to be analytic as defined in Sec. 15.4). If f (z) is entire (analytic

for all z; see Sec. 13.5), then (2) converges for all z. The functions z e , cos z, sin z, etc. have Maclaurin series, that is, Taylor series with center 0, similar to those in

calculus (Sec. 15.4).