- 3.2.1: For xn given by the following formulas, establish either the conver...
- 3.2.2: Give an example of two divergent sequences X and Y such that: ) (a)...
- 3.2.3: Show that if X and Y are sequences such that X and X + Y are conver...
- 3.2.4: Show that if X and Y are sequences such that X converges to x =1= 0...
- 3.2.5: Show that the following sequences are not convergent. (a) (2n), (b)...
- 3.2.6: Find the limits of the following sequences: (a) lim(2+1/n)2). (b) l...
- 3.2.7: If (b ) isaboundedsequence and lim(a ) = 0, show that lim(a b ) = o...
- 3.2.8: Explain why the result in equation (3) before Theorem 3.2.4 cannot ...
- 3.2.9: Let Yn := .Jm - Jii for n EN. Show that (Yn) and (JiiYn) converge. ...
- 3.2.10: Determine the following limits. (a) lim(3Jii)1/2n), (b) lim(n + l)l...
- 3.2.11: If 0 < a < b, determine lim an + bn ) . 1
- 3.2.12: If a > 0, b > 0, showthat lim (v'(n + a)(n + b) - n) = (a + b)/
- 3.2.13: U~ethe SqueezeTheorem3.2.7to determinethe limitsof the following. (...
- 3.2.14: Show that if zn := (an + bn)l/n where 0 < a < b, then lim(z n) = b....
- 3.2.15: Apply Theorem 3.2;11 to the following sequences, where a, b satisfy...
- 3.2.16: (a) Give an example of a convergent sequence (xn) of positive numbe...
- 3.2.17: Let X = (xn) be a sequenceof positivereal numberssuch that lim(xn+1...
- 3.2.18: Discuss the convergence of the following sequences, where a, b sati...
- 3.2.19: Let (xn) be a sequence of positive real numbers such that lim(x~/n)...
- 3.2.20: (a) Give an example of a convergent sequence (xn) of positive numbe...
- 3.2.21: (Thus, this property cannot be used as a test for convergence.) 2
- 3.2.22: Suppose that (xn) is a convergent sequence and (Yn) is such that fo...
- 3.2.23: Showthat if (xn)and (yn) are convergentsequences,then the sequences...
Solutions for Chapter 3.2: Limit Theorems
Full solutions for Introduction to Real Analysis | 3rd Edition
Addition property of inequality
If u < v , then u + w < v + w
A sample that sacrifices randomness for convenience
The gathering and processing of numerical information
Direction vector for a line
A vector in the direction of a line in three-dimensional space
The points (x, y, z) in space with x > 0 y > 0, and z > 0.
Grapher or graphing utility
Graphing calculator or a computer with graphing software.
Real numbers that are not rational, p. 2.
Logarithmic function with base b
The inverse of the exponential function y = bx, denoted by y = logb x
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.
The closest point to the Sun in a planet’s orbit.
Point-slope form (of a line)
y - y1 = m1x - x 12.
Power rule of logarithms
logb Rc = c logb R, R 7 0.
The formula x = -b 2b2 - 4ac2a used to solve ax 2 + bx + c = 0.
Rectangular coordinate system
See Cartesian coordinate system.
Solve by substitution
Method for solving systems of linear equations.
symmetric about the x-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
See Component form of a vector.
The y-value of the top of the viewing window.
The y-value of the bottom of the viewing window.