- 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 real number
Denoted by |a|, represents the number a or the positive number -a if a < 0.
A circular graphical display of categorical data
Difference of functions
(ƒ - g)(x) = ƒ(x) - g(x)
Equivalent systems of equations
Systems of equations that have the same solution.
Explicitly defined sequence
A sequence in which the kth term is given as a function of k.
The graph of the linear inequality y ? ax + b, y > ax + b y ? ax + b, or y < ax + b.
See Component form of a vector.
A shift of a graph to the left or right.
Imaginary part of a complex number
See Complex number.
Inverse reflection principle
If the graph of a relation is reflected across the line y = x , the graph of the inverse relation results.
Measure of center
A measure of the typical, middle, or average value for a data set
Natural logarithmic regression
A procedure for fitting a logarithmic curve to a set of data.
A set is ordered if it is possible to compare any two elements and say that one element is “less than” or “greater than” the other.
An angle in standard position whose terminal side lies on an axis.
A procedure for fitting a quadratic function to a set of data.
A sum where the interval is divided into n subintervals of equal length and is in the ith subinterval.
Sample standard deviation
The standard deviation computed using only a sample of the entire population.
Slope-intercept form (of a line)
y = mx + b
The line segment whose endpoints are the vertices of a hyperbola.
Usually the horizontal coordinate line in a Cartesian coordinate system with positive direction to the right,.
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