- 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
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
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)
A ratio equal to 1, used for unit conversion
The measure of an angle in degrees, minutes, and seconds
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.
See Linear regression line.
limx:aƒ1x2 = L means that ƒ(x) gets arbitrarily close to L as x gets arbitrarily close (but not equal) to a
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.
The number zero on a number line, or the point where the x- and y-axes cross in the Cartesian coordinate system, or the point where the x-, y-, and z-axes cross in Cartesian three-dimensional space
A set of parametric equations for a curve.
Product of complex numbers
(a + bi)(c + di) = (ac - bd) + (ad + bc)i
A square matrix with zero determinant
Transpose of a matrix
The matrix AT obtained by interchanging the rows and columns of A.
A shift of a graph up or down.
Usually the vertical coordinate line in a Cartesian coordinate system with positive direction up, pp. 12, 629.
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