- 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
A value ƒ(c) is an absolute minimum value of ƒ if ƒ(c) ? ƒ(x)for all x in the domain of ƒ.
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.
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)
A function of the form ƒ(x) = a ? bx,where ?0, b > 0 b ?1
The minimum, first quartile, median, third quartile, and maximum of a data set.
A function whose domain is the set of all natural numbers.
Connected subset of the real number line with at least two points, p. 4.
Limit to growth
See Logistic growth function.
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)
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.
Theorem In a right triangle with sides a and b and hypotenuse c, c2 = a2 + b2
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.
See Finite sequence, Infinite sequence.
Symmetric about the y-axis
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
The directed distance from the xy-plane to a point in space, or the third number in an ordered triple.
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