 3.6.1: Show that if (x,.) is an unbounded sequence, then there exists a pr...
 3.6.2: Give examples of properly divergent sequences (x,.) and (y,.) withY...
 3.6.3: Show that if x,. > 0 for all n eN, then lim(x,.) = 0 if and only if...
 3.6.4: Establish the proper divergence of the following sequences. (a) ( J...
 3.6.5: Is the sequence (n sin n) properly divergent?
 3.6.6: Let (x,.) be properly divergent and let (y,.) be such that lim(x,.y...
 3.6.7: Let (x,.) and (y,.) be sequences of positive numbers such that lim(...
 3.6.8: Investigate the convergence or the divergence of the following sequ...
 3.6.9: Let (x,.) and (y,.) be sequences of positive numbers such that lim(...
 3.6.10: Show that if lim(a,./n) = L, where L > 0, then lim(a,.) = +oo.
Solutions for Chapter 3.6: Properly Divergent Sequences
Full solutions for Introduction to Real Analysis  3rd Edition
ISBN: 9780471321484
Solutions for Chapter 3.6: Properly Divergent Sequences
Get Full SolutionsChapter 3.6: Properly Divergent Sequences includes 10 full stepbystep solutions. Introduction to Real Analysis was written by and is associated to the ISBN: 9780471321484. Since 10 problems in chapter 3.6: Properly Divergent Sequences have been answered, more than 2052 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Introduction to Real Analysis, edition: 3. This expansive textbook survival guide covers the following chapters and their solutions.

Absolute minimum
A value ƒ(c) is an absolute minimum value of ƒ if ƒ(c) ? ƒ(x)for all x in the domain of ƒ.

Categorical variable
In statistics, a nonnumerical variable such as gender or hair color. Numerical variables like zip codes, in which the numbers have no quantitative significance, are also considered to be categorical.

Common ratio
See Geometric sequence.

Decreasing on an interval
A function f is decreasing on an interval I if, for any two points in I, a positive change in x results in a negative change in ƒ(x)

Exponential function
A function of the form ƒ(x) = a ? bx,where ?0, b > 0 b ?1

Fivenumber summary
The minimum, first quartile, median, third quartile, and maximum of a data set.

Infinite sequence
A function whose domain is the set of all natural numbers.

Interval
Connected subset of the real number line with at least two points, p. 4.

Limit to growth
See Logistic growth function.

Mathematical induction
A process for proving that a statement is true for all natural numbers n by showing that it is true for n = 1 (the anchor) and that, if it is true for n = k, then it must be true for n = k + 1 (the inductive step)

Modified boxplot
A boxplot with the outliers removed.

Order of an m x n matrix
The order of an m x n matrix is m x n.

Power rule of logarithms
logb Rc = c logb R, R 7 0.

Pythagorean
Theorem In a right triangle with sides a and b and hypotenuse c, c2 = a2 + b2

Range screen
See Viewing window.

Removable discontinuity at x = a
lim x:a ƒ(x) = limx:a+ ƒ(x) but either the common limit is not equal ƒ(a) to ƒ(a) or is not defined

Root of a number
See Principal nth root.

Sequence
See Finite sequence, Infinite sequence.

Symmetric about the yaxis
A graph in which (x, y) is on the graph whenever (x, y) is; or a graph in which (r, ?) or (r, ?, ?) is on the graph whenever (r, ?) is

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