- 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 Cauchy-HadamardTheorem9.4.9.
- 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) :.: e-1/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 e-t 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
Annual percentage rate (APR)
The annual interest rate
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
A coordinate plane used to represent the complex numbers. The x-axis of the complex plane is called the real axis and the y-axis is the imaginary axis
Composition of functions
(f ? g) (x) = f (g(x))
A ratio equal to 1, used for unit conversion
The left side of u(v + w) = uv + uw.
A relation that associates each value in the domain with exactly one value in the range.
The numbers . . ., -3, -2, -1, 0,1,2,...2
See Polynomial function in x
A function that can be written in the form ƒ(x) = mx + b, where and b are real numbers
A logarithm with base e.
See Linear programming problem.
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).
The principle of experimental design that makes it possible to use the laws of probability when making inferences.
See Viewing window.
See Division algorithm for polynomials.
A 90° angle.
A square matrix with zero determinant
Sum of complex numbers
(a + bi) + (c + di) = (a + c) + (b + d)i
The directed distance from the xy-plane to a point in space, or the third number in an ordered triple.