 9.4.1: Discuss the convergence and the uniform convergence of the series L...
 9.4.2: If L an is an absolutely convergent series, then the series Lan sin...
 9.4.3: Let (en) be a decreasing sequence of positive numbers. If L en sin ...
 9.4.4: Discuss the cases R = 0, R = +00 in the CauchyHadamardTheorem9.4.9.
 9.4.5: Show that the radius of convergence R of the power series L anxn is...
 9.4.6: Determine the radius of convergence of the series L anxn, where an ...
 9.4.7: If an := 1 when n is the Square of a natural number and an := 0 oth...
 9.4.8: Prove in detail that 1imsup(lnan Il/n) = lim sup(lan Il/n).
 9.4.9: If 0 < p ~ IanI ~ q for all n E N, find the radius of convergence o...
 9.4.10: Let I(x) = LanXn for Ixl < R. If I(x) = I(x) for alllxl < R, show ...
 9.4.11: Prove that if I is defined for IxI < r and if there exists a consta...
 9.4.12: Prove by Induction that the function given by f(x) :.: e1/x2 for x...
 9.4.13: Give an example of a function which is equal to its Taylor series e...
 9.4.14: Use the Lagrange fonn of the remainder to justify the general Binom...
 9.4.15: (Geometric series) Show directly that if IxI < 1, then 1/(1  x) = ...
 9.4.16: Showby integratingthe seriesfor 1/(1 + x) that if Ixl < 1, then(X) ...
 9.4.17: Show that if Ixl < 1, then Arctan x = nL=O2n + x2n+l.
 9.4.18: Show that if Ixl < 1, then Arcsin x =nL=O 2 .4. 2n . 2n + 1.
 9.4.19: Find a series expansion for 0 et dt for x e IR.
 9.4.20: If exe IRand Ikl < 1, the integral F(ex,k) := la(1  k2(sinx)2r'/2 ...
Solutions for Chapter 9.4: Series of Functions
Full solutions for Introduction to Real Analysis  3rd Edition
ISBN: 9780471321484
Solutions for Chapter 9.4: Series of Functions
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Introduction to Real Analysis was written by and is associated to the ISBN: 9780471321484. Since 20 problems in chapter 9.4: Series of Functions have been answered, more than 3155 students have viewed full stepbystep solutions from this chapter. Chapter 9.4: Series of Functions includes 20 full stepbystep solutions. This textbook survival guide was created for the textbook: Introduction to Real Analysis, edition: 3.

Complex plane
A coordinate plane used to represent the complex numbers. The xaxis of the complex plane is called the real axis and the yaxis is the imaginary axis

Constant
A letter or symbol that stands for a specific number,

Constant function (on an interval)
ƒ(x 1) = ƒ(x 2) x for any x1 and x2 (in the interval)

Conversion factor
A ratio equal to 1, used for unit conversion

DMS measure
The measure of an angle in degrees, minutes, and seconds

Doubleblind experiment
A blind experiment in which the researcher gathering data from the subjects is not told which subjects have received which treatment

Equal complex numbers
Complex numbers whose real parts are equal and whose imaginary parts are equal.

Exponential growth function
Growth modeled by ƒ(x) = a ? b a > 0, b > 1 .

Imaginary part of a complex number
See Complex number.

Leastsquares line
See Linear regression line.

Limit
limx:aƒ1x2 = L means that ƒ(x) gets arbitrarily close to L as x gets arbitrarily close (but not equal) to a

Matrix element
Any of the real numbers in a matrix

nth power of a
The number with n factors of a , where n is the exponent and a is the base.

Origin
The number zero on a number line, or the point where the x and yaxes cross in the Cartesian coordinate system, or the point where the x, y, and zaxes cross in Cartesian threedimensional space

Parametrization
A set of parametric equations for a curve.

Product of complex numbers
(a + bi)(c + di) = (ac  bd) + (ad + bc)i

Singular matrix
A square matrix with zero determinant

Transpose of a matrix
The matrix AT obtained by interchanging the rows and columns of A.

Vertical translation
A shift of a graph up or down.

yaxis
Usually the vertical coordinate line in a Cartesian coordinate system with positive direction up, pp. 12, 629.
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