Limit of a Sequence
- [Play
Video] 4½ minutes: Algebraic View of Limits. Example
involving sums and quotients.
- [Play
Video] 5½ minutes: Limits and Error Control for Linear
Expressions
- [Play
Video] 2¾ minutes: Error Control to N decimal Places, say 5 or
10.
- [Play
Video] 3¼ minutes: Limits as Error Control for an
unlimited number of decimal places.
Suppose g(n) is a function of whole numbers n > 0.
Then g(1),g(2),g(3),¼, form an infinite sequence
of points. This sequence is said to converge to a finite limit
if and only if there is a real number L such that
for every positive number e = [1/2] [1/(10k)] > 0 there is an N such that
|
n > N implies |g(n)-L| < e = |
1
2 |
|
1
10k |
|
|
In the latter case, a limit L is said to exist and we
write
The decimal-free equivalent form of
the foregoing definition would relax the requirement that
e = [1/2] [1/(10k)].
The precise decimal-based definition of a Cauchy sequence g(n)
is as follows.
For every whole number k > 0, there exist
a whole number N such that
|
n ³ N and m ³ N implies |g(n)-g(m)| £ e = |
1
2 |
|
1
10k |
· |
|
The equivalent decimal-free description or
definition of a Cauchy-Sequence g(n) is given next.
For
every positive real number e > 0, there exist a
whole number N such tha
|
n ³ N and m ³ N implies |g(n)-g(m)| £ e. |
|
| |
Why Slopes
and
More Math
understanding & explaining
Reason and Math
Volume 3
Printed in Canada
ISBN 0-9697564-3-7
|
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Content Guide
Foreword
Chapter Descriptions
1. Introduction
Preview -why slopes in 1983
2. Second Preview Begins
2 Skier in Motion (V)
2 The Skier (V)
2. Position Dependent (V)
3 Slope & Extrema (V)
4 Single Factor Analysis (V)
4 Two Factor (V)
4 More Factors (V)
4 With Divisors (V)
5 Maxima & Minima Tests
6 Jumps & Discontinuities
8 Review (optional)
9 On Calculus Studies
11 Slope of Slope
13 Acceleration
14 Limits & Error Control (V)
14 Limit of a Fn.
14. Limited Error Control
14 Signif. Digits
14 Cauchy Limits
14 Sequence Limits
14 Decimal Arith.
15 What is Slope (V)
15 Slope Calculation (V)
15 Slope, a Limit
15 Tangent Lines
15 Linear Approx.,
15 Limits via Algebra (V)
15 Recap.
PS.Chain Rule for Polys
PS Chain Rule- General (V) -
PS More Chain Rule (V)
PS - Sign Analysis (V)
16 What is Velocity
17 What is Area
18 Integration
18 Area Calculation
18 Fn DefN, 6 Ways
19 Logs & Powers
19 Natural Log.
19 Exponential Fn.
20 What's Next
21 Add Vectors
22 Complex #'s
23 Complex #'s
23 Trig Identity
23 Proofs of.
24 Complex Logs etc
Units in Calculations:
7 Velocity
7 Varying Velocity Example
7. Velocity Calculation
7 Changing Units
7 Same Velocity Motions
10 Slopes without Units.
10 Units & Slopes
10 Units in Cost vs. Quantity
10 How Units Appear
10 Unit Elimination
10 Partial Elimination
10 Interest & Units
12 More on Units
Enriched material: The Appendices of Volume
3 are located in the Real
Analysis Area.
Pigeon Hole Principle
Constant Difference Thm
Continuous Functions
Rational Functions
Mean Value Theorem
One Side Range Theorem
Range On One Side
Theorem
Integration
& Lipschitz
Continuity
These appendices continue the
decimal viewpoint of limits, error
control and continuity begun
in Chapter 14. The One Sided
Range Theorem is a postscript,
not in printed version.
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