CAS PROJECT. Sequences and Series. (a) Write a program for

Are the following sequences Zl, Z2, ... , Zn> ... bounded? Convergent? Find their limit points. (Show the details of your work.)

Are the following sequences Zl, Z2, ... , Zn> ... bounded? Convergent? Find their limit points. (Show the details of your work.)

Are the following sequences Zl, Z2, ... , Zn> ... bounded? Convergent? Find their limit points. (Show the details of your work.)

Are the following sequences Zl, Z2, ... , Zn> ... bounded? Convergent? Find their limit points. (Show the details of your work.)

Are the following sequences Zl, Z2, ... , Zn> ... bounded? Convergent? Find their limit points. (Show the details of your work.)

Are the following sequences Zl, Z2, ... , Zn> ... bounded? Convergent? Find their limit points. (Show the details of your work.)

Are the following sequences Zl, Z2, ... , Zn> ... bounded? Convergent? Find their limit points. (Show the details of your work.)

Are the following sequences Zl, Z2, ... , Zn> ... bounded? Convergent? Find their limit points. (Show the details of your work.)

Are the following sequences Zl, Z2, ... , Zn> ... bounded? Convergent? Find their limit points. (Show the details of your work.)

Are the following sequences Zl, Z2, ... , Zn> ... bounded? Convergent? Find their limit points. (Show the details of your work.)

Illustrate Theorem 1 by an example of your own.

(Uniqueness of limit) Show that if a sequence converges. its limit is unique.

(Addition) If ZI, Z2, ... converges with the limit [and ZI *, Z2 *, ... converges with the limit [*, show that Zl + Zl *':::2 + Z2*' ... converges with the limit [ + [*.

(Multiplication) Show that under the assumptions of Prob. L3 the sequence ZlZ1*' Z2Z2*' ... converges with the limit U*

(Boundedness) Show that a complex sequence is bounded if and onl y if the two corresponding sequences of the real parts and of the imaginary parts are bounded.

Are the following series convergent or divergent? (Give a reason.)6.2:----

Are the following series convergent or divergent? (Give a reason.)~o (2n + I)!

Are the following series convergent or divergent? (Give a reason.) :L -2- 1 --. n - 21

Are the following series convergent or divergent? (Give a reason.)2: Vn n~l n

Are the following series convergent or divergent? (Give a reason.).2: n=2 In n

Are the following series convergent or divergent? (Give a reason.)

Are the following series convergent or divergent? (Give a reason.)2: -'- (1 + on

Are the following series convergent or divergent? (Give a reason.).2: 3n + 2i

Are the following series convergent or divergent? (Give a reason.)

What is the difference between (7) and just stating IZn+l/Znl < I?

Illustrate Theorem 2 by an example of your choice.

For what n do we obtain the term of greatest absolute value of the series in Example 4? About how big is it? First guess, then calculate it by the Stirling formula in Sec. 24.4.

Give another example showing that Theorem 7 is more general than Theorem 8.

CAS PROJECT. Sequences and Series. (a) Write a program for graphing complex sequences. Apply it to sequences of your choice that have interesting "geometrical" properties (e.g., lying on an ellipse, spiraling toward its limit, etc.). (b) Write a program for computing and graphing numeric values of the first n partial sums of a series of complex numbers. Use the program to experiment with the rapidity of convergence of series of your choice.

TEAM PROJECT. Series. ta) Absolute convergence. Show that if a series converges absolutely, it is convergent. (b) Write a short report on the basic concepts and properties of series of numbers, explaining in each case whether or not they carry over from real series (discussed in calculus) to complex series, with reasons given (c) Estimate of the remainder. Let Izn+llznl ~ q < 1, so that the series Zl + Z2 + ... converges by the ratio test. Show that the remainder Rn = Zn+l + Zn+2 + ... satisfies the inequality IRnl ~ Izn+ll/(I - q). (d) Using (c), find how many terms suffice for computing the sum s of the series with an error not exceeding 0.05 and compute s to this accuracy. (e) Find other applications of the estimate in (c)

(Powers missing) Show that if ~ a"z'YI has radius of convergence R (assumed finite), then ~ a,,:!''' has radius of convergence "\'R. Give examples

(Convergence behavior) Illustrate the facts shown by Examples 1-3 by further examples of your own.

Find the center and the radius of convergence of the following power series. (Show the details.)

Find the center and the radius of convergence of the following power series. (Show the details.)

Find the center and the radius of convergence of the following power series. (Show the details.)

Find the center and the radius of convergence of the following power series. (Show the details.)

Find the center and the radius of convergence of the following power series. (Show the details.)

Find the center and the radius of convergence of the following power series. (Show the details.)

Find the center and the radius of convergence of the following power series. (Show the details.)

Find the center and the radius of convergence of the following power series. (Show the details.)

Find the center and the radius of convergence of the following power series. (Show the details.)

Find the center and the radius of convergence of the following power series. (Show the details.)

Find the center and the radius of convergence of the following power series. (Show the details.)

Find the center and the radius of convergence of the following power series. (Show the details.)

Find the center and the radius of convergence of the following power series. (Show the details.)

Find the center and the radius of convergence of the following power series. (Show the details.)

Find the center and the radius of convergence of the following power series. (Show the details.)

Find the center and the radius of convergence of the following power series. (Show the details.)

CAS PROJECT. Radius of Convergence. Write a program for computing R from (6), (6*), or (6"'*). in this order, depending on the existence of the limits needed. Test the program on series of your choice and such that all three formulas (6). (6*), and (6**) will come up.

TEAM PROJECT. Radius of Convergence. (a) Formula (6) for R contains iOn/On+li, not iOn+l/Oni. How could you memorize this by using a qualitative argument?(b) Change of coefficients. What happens to R (0 < R < 00) if you (i) multiply all On by k * O. (ii) multiply On by k fl * O. (iii) replace On by lion? (c) Example 6 extends Theorem 2 to nonconvergent cases of O .. /On+l' Do you understand the principle of "mixing" by which Example 6 was obtained? Use this principle for making up further examples. (d) Does there exist a power series in powers of z that converges at z = 30 + 10i and diverges at z = 31 - 6i? (Give reason.)

Find the radius of convergence in two ways: (a) directly by the Cauchy-Hadamard formula in Sec. 15.2. (b) from a series of simpler telms by using Theorem 3 or Theorem 4.

Find the radius of convergence in two ways: (a) directly by the Cauchy-Hadamard formula in Sec. 15.2. (b) from a series of simpler telms by using Theorem 3 or Theorem 4.

Find the radius of convergence in two ways: (a) directly by the Cauchy-Hadamard formula in Sec. 15.2. (b) from a series of simpler telms by using Theorem 3 or Theorem 4.

Find the radius of convergence in two ways: (a) directly by the Cauchy-Hadamard formula in Sec. 15.2. (b) from a series of simpler telms by using Theorem 3 or Theorem 4.

Find the radius of convergence in two ways: (a) directly by the Cauchy-Hadamard formula in Sec. 15.2. (b) from a series of simpler telms by using Theorem 3 or Theorem 4.

Find the radius of convergence in two ways: (a) directly by the Cauchy-Hadamard formula in Sec. 15.2. (b) from a series of simpler telms by using Theorem 3 or Theorem 4.

Find the radius of convergence in two ways: (a) directly by the Cauchy-Hadamard formula in Sec. 15.2. (b) from a series of simpler telms by using Theorem 3 or Theorem 4.

Find the radius of convergence in two ways: (a) directly by the Cauchy-Hadamard formula in Sec. 15.2. (b) from a series of simpler telms by using Theorem 3 or Theorem 4.

Find the radius of convergence in two ways: (a) directly by the Cauchy-Hadamard formula in Sec. 15.2. (b) from a series of simpler telms by using Theorem 3 or Theorem 4.

Find the radius of convergence in two ways: (a) directly by the Cauchy-Hadamard formula in Sec. 15.2. (b) from a series of simpler telms by using Theorem 3 or Theorem 4.

(Addition and subtraction) Write our the details of the proof on terrnwise addition and subtraction of power series.

(Cauchy product) Show that (1 - Z)-2 = L';;~O (n + l)zn tal by using the Cauchy product, (b) by differentiating a suitable series.

(Cauchy product) Show that the Cauchy product of L~~O zn/n! multiplied by itself gives L~~O (2zyn/n!.

(On Theorem 3) Prove that Vn ~ I as n ~ ex; (as claimed in the proof of Theorem 3).

(On Theorems 3 and 4) Find further examples of your own.

State clearly and explicitly where and how you are using Theorem 2.(Bionomial coefficients) Using (1 + z)P(J + z)q = (1 + z)p+q. obtain the basic relation

State clearly and explicitly where and how you are using Theorem 2. (Odd function) If .f(z) in (1) is odd (i.e., .f(-z) = -.f(z, show that an = 0 for even n. Give examples

State clearly and explicitly where and how you are using Theorem 2.(Even functions) If .f(z) in (1) is even (i.e., .f( - z) = .f(z, show that an = 0 for odd n. Give examples.

State clearly and explicitly where and how you are using Theorem 2. Find applications of Theorem 2 in differential equations and elsewhere

State clearly and explicitly where and how you are using Theorem 2.TEAM PROJECT. Fibonacci nmnbers.2 tal The Fibonacci numbers are recursively defined by ao = al = 1. an +l = an + an-l if n = 1. 2 ..... Find the limit of the sequence (an+l/an)' (b) Fibonacci's rabbit problem. Compute a list of a1 . .... a12' Show that a12 = 233 is the munber of pairs of rabbits after l2 months if initially there is 1 pair and each pair generates I pair per month, beginning in the second month of existence (no deaths occuning). (c) Generating function. Show that the generating junction of the Fibonacci numbers is .f(z) = I/(1 - z - Z2); that is, if a power series (l) represents this .f(z), its coefficients must be the Fibonacci numbers and conversely. Hint. Start from .f(z) (1 - z - Z2) = I and use Theorem 2.

Find the Taylor or Maclaurin series of the given function with the given point as center and detennine the radius of convergence.

Find the Taylor or Maclaurin series of the given function with the given point as center and detennine the radius of convergence.

Find the Taylor or Maclaurin series of the given function with the given point as center and detennine the radius of convergence.

Find the Taylor or Maclaurin series of the given function with the given point as center and detennine the radius of convergence.

Find the Taylor or Maclaurin series of the given function with the given point as center and detennine the radius of convergence.

Find the Taylor or Maclaurin series of the given function with the given point as center and detennine the radius of convergence.

Find the Taylor or Maclaurin series of the given function with the given point as center and detennine the radius of convergence.

Find the Taylor or Maclaurin series of the given function with the given point as center and detennine the radius of convergence.

Find the Taylor or Maclaurin series of the given function with the given point as center and detennine the radius of convergence.

Find the Taylor or Maclaurin series of the given function with the given point as center and detennine the radius of convergence.

Find the Taylor or Maclaurin series of the given function with the given point as center and detennine the radius of convergence.

Find the Taylor or Maclaurin series of the given function with the given point as center and detennine the radius of convergence.

Find the Maclaurin series by tennwise integrating the integrand. (The integrals cannot be evaluated by the usual methods of calculus. They define the error function erf z, sine integral Si(z). and Fresnel integrals4 S(z) and C(z). which occur in statistics, heat conduction. optics, and other applications. These are special so-called higher transcendental functions.)

Find the Maclaurin series by tennwise integrating the integrand. (The integrals cannot be evaluated by the usual methods of calculus. They define the error function erf z, sine integral Si(z). and Fresnel integrals4 S(z) and C(z). which occur in statistics, heat conduction. optics, and other applications. These are special so-called higher transcendental functions.)

Find the Maclaurin series by tennwise integrating the integrand. (The integrals cannot be evaluated by the usual methods of calculus. They define the error function erf z, sine integral Si(z). and Fresnel integrals4 S(z) and C(z). which occur in statistics, heat conduction. optics, and other applications. These are special so-called higher transcendental functions.)

Find the Maclaurin series by tennwise integrating the integrand. (The integrals cannot be evaluated by the usual methods of calculus. They define the error function erf z, sine integral Si(z). and Fresnel integrals4 S(z) and C(z). which occur in statistics, heat conduction. optics, and other applications. These are special so-called higher transcendental functions.)

CAS PROJECT. sec, tan, arcsin. (a) Euler numbers. The Maclaurin series E22 E44 (21) sec z = Eo - - z + - z - + ... 2! 4! defines the Elller numbers E2n- Show that Eo = 1, E2 = -I, E4 = 5, E6 = -61. Write a program that computes the E2n from the coefficient formula in (1) or extracts them as a list from the series. (For tables see Ref. [GRI]. p. 810. listed in App. 1.) (b) Bernoulli numbers. The Maclaurin series defines the Bernoulli numbers Bn. Using undetermined coefficients, show that I Bl = , B2 = - B3 = O. (23) 2 6 I 1 B4 = -- B5 = 0, B6 = - 30 42 Write a program for computing Bn. (c) Tangent. Using (1), (2), Sec. 13.6, and (22), show that tan z has the following Maclaurin series and calculate from it a table of Bo, ... , B2O: 2i 4

(Inverse sine) Developing uV I - Z2 and integrating, show that arcsin z = z + () ~ + (~:!) ~ + (~) _7 +. 2'4' 6 7 (izl < 1). Show that this series represents the principal value of arcsin z (defined in Team Project 30. Sec. 13.7).

(Undetennined coefficients) Using the relation sin z = tan Z cos Z and the Maclaurin series of sin z and cos z, find the first four nonzero terms of the Maclaurin series of tan z. (Show the details.)

TEAM PROJECT. Properties from Maclaurin Series. Clearly, from series we can compute function values. In this project we show that properties of functions can often be discovered from their Taylor or Maclaurin series. Using suitable series, prove the following. (a) The fonnulas for the derivatives of e2 , cos z, sin z, cosh Z, sinh z, and Ln (1 + z) (b) 4(iZ + e-iz) = cos Z (c) sin z =1= 0 for all pure imaginalY Z = iy *" 0

Prove that the given series converges uniformly In the indicated region.

Prove that the given series converges uniformly In the indicated region.

Prove that the given series converges uniformly In the indicated region.

Prove that the given series converges uniformly In the indicated region.

Prove that the given series converges uniformly In the indicated region.

Prove that the given series converges uniformly In the indicated region.

Prove that the given series converges uniformly In the indicated region.

Prove that the given series converges uniformly In the indicated region.

Find the region of uniform convergence. (Give reason.)

Find the region of uniform convergence. (Give reason.)

Find the region of uniform convergence. (Give reason.)

Find the region of uniform convergence. (Give reason.)

Find the region of uniform convergence. (Give reason.)

Find the region of uniform convergence. (Give reason.)

Find the region of uniform convergence. (Give reason.)

Find the region of uniform convergence. (Give reason.)

CAS PROJECT. Graphs of Partial Sums. (a) Figure 365. Produce this exciting figure using your software and adding fm1her curves. say, those of S256' SI024' etc. (b) Power series. Study the nonuniformity of convergence experimentally by plotting partial sums near the endpoints of the convergence interval for real z = x.

TEAM PROJECT. Uniform Convergence. (a) Weierstrass M-test. Give a proof. (b) Termwise differentiation. Derive Theorem 4 from Theorem 3. (c) Subregions. Prove that uniform convergence of a series in a region C implies unifonn convergence in any portion of C. Is the converse true? (d) Example 2. Find the precise region of convergence of the series in Example 2 with x replaced by a complex variable z. (e) Figure 366. Show that x 2 ~;;'~1 (1 + x2 )-m = 1 if x =F 0 and 0 if x = O. Verify hy computation that the partial sums .1'10 S2' S3, look as shown in Fig. 366. -1 y 1 o s Fig. 366. Sum 5 and partial sums III Team Project 18(e)

Show that (9) in Sec. 12.5 with coefficients (10) is a solution of the heat equation for t > 0, assuming that f(x) is continuous on the interval 0 ;: x ;: L and has one-sided derivatives at all interior points of that interval. Proceed as follows. Show that IB"I is bounded, say IB"I < K for all n. Conclude that if t ~ to > 0 and. by the Weierstrass test. the series (9) converges uniformly with respect to x and t for f ~ fo, 0 ;: x ;: L. Using Theorem 2. show that II(X, t) is continuous for 1 ~ 10 and thus satisfies the boundary conditions (2) for f ~ fo.

Show that (9) in Sec. 12.5 with coefficients (10) is a solution of the heat equation for t > 0, assuming that f(x) is continuous on the interval 0 ;: x ;: L and has one-sided derivatives at all interior points of that interval. Proceed as follows. Show that Iillln/iltl < An2 Ke-An2to if 1 ~ to and the series of the expressions on the right converges. by the ratio test. Conclude from this. the Weierstrass test, and Theorem 4 that the series (9) can be differentiated term by term with respect to t and the resulting series has the sum duliN. Show that (9) can be differentiated twice with respect to x and the resulting series has the sum a2u/ilx2 . Conclude from this and the result to Prob. 19 that (9) is a solution of the heat equation for all t ~ to. (The proof that (9) satisfies the given initial condition can be found in Ref. [CIO] listed in App. 1.)

What are power series? Why are these series very important in complex analysis?

State from memory the ratio test, the root test, and the Cauchy-Hadamard fomlLlla for the radius of convergence.

What is absolute convergence? Conditional convergence? Uniform convergence?

What do you know about the convergence of power series?

What is a Taylor series? What was the idea of obtaining it from Cauchy's integral formula?

Give examples of practical methods for obtaining Taylor series.

What have power series to do with analytic functions?

Can propel1ies of functions be discovered from their Maclaurin series? If so, give examples.

Make a list of Maclaurin series of c. cos z. sin z, cosh z, sinh z, Ln (1 - z) from memory.

What do you know about adding and multiplying power series?

Find the radius of convergence. Can you identify the sum as a familiar function in some of the problems? (Show the details of your work.) L...-- n!

Find the radius of convergence. Can you identify the sum as a familiar function in some of the problems? (Show the details of your work.)

Find the radius of convergence. Can you identify the sum as a familiar function in some of the problems? (Show the details of your work.) L n~O 2n + 1

Find the radius of convergence. Can you identify the sum as a familiar function in some of the problems? (Show the details of your work.) L (-I)n zn n~O (2n)!

Find the radius of convergence. Can you identify the sum as a familiar function in some of the problems? (Show the details of your work.)

Find the radius of convergence. Can you identify the sum as a familiar function in some of the problems? (Show the details of your work.) L (z - 2y2n+l n~O (2n + I)!

Find the radius of convergence. Can you identify the sum as a familiar function in some of the problems? (Show the details of your work.)

Find the radius of convergence. Can you identify the sum as a familiar function in some of the problems? (Show the details of your work.) L -- n~O (217)!

Find the radius of convergence. Can you identify the sum as a familiar function in some of the problems? (Show the details of your work.)

Find the radius of convergence. Can you identify the sum as a familiar function in some of the problems? (Show the details of your work.) L n~O (3 + 4i)n

Find the Taylor or Maclaurin series with the given point as center and determine the radius of convergence. (Show details.) e". 7ri

Find the Taylor or Maclaurin series with the given point as center and determine the radius of convergence. (Show details.)Ln z. 2

Find the Taylor or Maclaurin series with the given point as center and determine the radius of convergence. (Show details.) 1/(1 - z), -1

Find the Taylor or Maclaurin series with the given point as center and determine the radius of convergence. (Show details.)11(4 - 3z), 1 + i

Find the Taylor or Maclaurin series with the given point as center and determine the radius of convergence. (Show details.)11(1 - d, 0

Find the Taylor or Maclaurin series with the given point as center and determine the radius of convergence. (Show details.)l1z2, i

Find the Taylor or Maclaurin series with the given point as center and determine the radius of convergence. (Show details.) liz, -i

Find the Taylor or Maclaurin series with the given point as center and determine the radius of convergence. (Show details.)I"t- 1(et - 1) dt, 0

Find the Taylor or Maclaurin series with the given point as center and determine the radius of convergence. (Show details.) cos Z, ~7r

Find the Taylor or Maclaurin series with the given point as center and determine the radius of convergence. (Show details.)sin2 z, 0

Does every function fez) have a Taylor series?

Does there exist a Taylor series in powers of z - 1 - i that diverges at 5 + 5i but converges at 4 + 6i?

Do we obtain an analytic function if we replace x by z in the Maclaurin series of a real function f(x)?

Using Maclaurin series. show that if fez) is even. its integral (with a suitable constant of integration) is odd.

Obtain the first few terms of the Maclaurin series of tan z by using the Cauchy product and sin z = cos z tan z.