- 9.3.1: Test the following series for convergence and for absolute converge...
- 9.3.2: If sn is the nth partial sum of the alternating series L~l (_1)n+l ...
- 9.3.3: Give an example to show that the Alternating Series Test 9.3.2 may ...
- 9.3.4: Show that the Alternating Series Test is a consequence of Dirichlet...
- 9.3.5: Consider the series1 1 1 1 1 11 +-+ ++--...2 345 67'where the signs...
- 9.3.6: Let an e JRfor n e N and let p < q. If the seriesL anInPis converge...
- 9.3.7: If p and q are positive numbers, show that L ( _1)n (In n)P Inq is ...
- 9.3.8: Discuss the series whose nth term is:nn nn(a) (_1)n. Hn+l' (b) _ _'...
- 9.3.9: If the partial sums of L an are bounded, show that the series L~l a...
- 9.3.10: If the partial sums sn of L~l an are bounded, show that the series ...
- 9.3.11: Can Dirichlet's Test be applied to establish the convergence of1 1 ...
- 9.3.12: Show that the hypotheses that the sequence X := (xn) is decreasing ...
- 9.3.13: If (an) is a bounded decreasing sequence and (bn) is a bounded incr...
- 9.3.14: Show that if the partial sums sn of the series L~l ak satisfy ISnI ...
- 9.3.15: Suppose that L an is a convergent series of real numbers. Either pr...
Solutions for Chapter 9.3: Tests for Nonabsolute Convergence
Full solutions for Introduction to Real Analysis | 3rd Edition
Absolute value of a vector
See Magnitude of a vector.
A combination of variables and constants involving addition, subtraction, multiplication, division, powers, and roots
A function is bounded above if there is a number B such that ƒ(x) ? B for all x in the domain of ƒ.
An expression a + bi, where a (the real part) and b (the imaginary part) are real numbers
Expanded form of a series
A series written explicitly as a sum of terms (not in summation notation).
A function of the form ƒ(x) = a ? bx,where ?0, b > 0 b ?1
First-degree equation in x , y, and z
An equation that can be written in the form.
Limit at infinity
limx: qƒ1x2 = L means that ƒ1x2 gets arbitrarily close to L as x gets arbitrarily large; lim x:- q ƒ1x2 means that gets arbitrarily close to L as gets arbitrarily large
Lower bound for real zeros
A number c is a lower bound for the set of real zeros of ƒ if ƒ(x) Z 0 whenever x < c
Multiplication property of equality
If u = v and w = z, then uw = vz
A function in which ƒ(x)is a polynomial in x, p. 158.
Quotient of functions
a ƒ g b(x) = ƒ(x) g(x) , g(x) ? 0
Real part of a complex number
See Complex number.
A real number.
A positive number written as c x 10m, where 1 ? c < 10 and m is an integer.
Sum of functions
(ƒ + g)(x) = ƒ(x) + g(x)
Terminal side of an angle
A letter that represents an unspecified number.
See Power function.
A shift of a graph up or down.