 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 9581 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.

Annual percentage rate (APR)
The annual interest rate

Combinatorics
A branch of mathematics related to determining the number of elements of a set or the number of ways objects can be arranged or combined

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

Composition of functions
(f ? g) (x) = f (g(x))

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

Factored form
The left side of u(v + w) = uv + uw.

Function
A relation that associates each value in the domain with exactly one value in the range.

Integers
The numbers . . ., 3, 2, 1, 0,1,2,...2

Leading coefficient
See Polynomial function in x

Linear function
A function that can be written in the form ƒ(x) = mx + b, where and b are real numbers

Natural logarithm
A logarithm with base e.

Objective function
See Linear programming problem.

Parabola
The graph of a quadratic function, or the set of points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).

Randomization
The principle of experimental design that makes it possible to use the laws of probability when making inferences.

Range screen
See Viewing window.

Remainder polynomial
See Division algorithm for polynomials.

Right angle
A 90° angle.

Singular matrix
A square matrix with zero determinant

Sum of complex numbers
(a + bi) + (c + di) = (a + c) + (b + d)i

zcoordinate
The directed distance from the xyplane to a point in space, or the third number in an ordered triple.