Advanced Calculus Single Variable

8.6.1 Interest Compounded Continuously

Suppose you put money in the bank and it accrues interest at the rate of r per payment period. These terms need a little explanation. If the payment period is one month, and you started with $100 then the amount at the end of one month would equal 100

= 100 + 100r. In this the second term is the interest and the first is called the principal. Now you have 100
in the bank. This becomes the new principal. How much will you have at the end of the second month? By analogy to what was just done it would equal

In general, the amount you would have at the end of n months is 100

n.

When a bank says they offer 6% compounded monthly, this means r, the rate per payment period equals .06∕12. Consider the problem of a rate of r per year and compounding the interest n times a year and letting n increase without bound. This is what is meant by compounding continuously. The interest rate per payment period is then r∕n and the number of payment periods after time t years is approximately tn. From the above the amount in the account after t years is

(8.30)

Recall from Example 8.6.10 that limy→∞

y = ex. The expression in 8.30 can be written as

and so, taking the limit as n →∞, you get

This shows how to compound interest continuously.

Example 8.6.11 Suppose you have $100 and you put it in a savings account which pays 6% compounded continuously. How much will you have at the end of 4 years?

From the above discussion, this would be 100e

4 = 127.12. Thus, in 4 years, you would gain interest of about $27.

Chapter August 25, 2018 1 Introduction

Chapter 2 The Real And Complex Numbers

2.1 Real and Rational Numbers
2.2 Exercises
2.3 Set Notation
2.4 Order
2.5 Exercises
2.6 The Binomial Theorem
2.7 Well Ordering Principle And Archimedean Property
2.8 Arithmetic of Integers
2.9 Exercises
2.10 Completeness of ℝ
2.11 Existence Of Roots
2.12 Exercises
2.13 The Complex Numbers
2.14 The Cauchy Schwarz Inequality
2.15 Integer Multiples of Irrational Numbers
2.16 Exercises

Chapter 3 Set Theory

3.1 Basic Definitions
3.2 The Schroder Bernstein Theorem
3.3 Equivalence Relations
3.4 Exercises

Chapter 4 Functions And Sequences

4.1 General Considerations
4.2 Sequences
4.3 Exercises
4.4 The Limit Of A Sequence
4.5 The Nested Interval Lemma
4.6 Exercises
4.7 Compactness
4.7.1 Sequential Compactness
4.7.2 Closed And Open Sets
4.7.3 Compactness And Open Coverings
4.8 Exercises
4.9 Cauchy Sequences And Completeness
4.9.1 Decimals
4.9.2 lim sup and lim inf
4.9.3 Shrinking Diameters
4.10 Exercises

Chapter 5 Infinite Series Of Numbers

5.1 Basic Considerations
5.1.1 Absolute Convergence
5.2 Exercises
5.3 More Tests For Convergence
5.3.1 Convergence Because Of Cancellation
5.3.2 Ratio And Root Tests
5.4 Double Series
5.5 Exercises

Chapter 6 Continuous Functions

6.1 Equivalent Formulations Of Continuity
6.2 Exercises
6.3 The Extreme Values Theorem
6.4 The Intermediate Value Theorem
6.5 Connected Sets
6.6 Exercises
6.7 Uniform Continuity
6.8 Exercises
6.9 Sequences And Series Of Functions
6.10 Sequences Of Polynomials, Weierstrass Approximation
6.11 Ascoli Arzela Theorem∗
6.12 Exercises

Chapter 7 The Derivative

7.1 Limit Of A Function
7.2 Exercises
7.3 The Definition Of The Derivative
7.4 Continuous And Nowhere Differentiable
7.5 Finding The Derivative
7.6 Local Extreme Points
7.7 Exercises
7.8 Mean Value Theorem
7.9 Exercises
7.10 Derivatives Of Inverse Functions
7.11 Derivatives And Limits Of Sequences
7.12 Exercises

Chapter 8 Power Series

8.1 Functions Defined In Terms Of Series
8.2 Operations On Power Series
8.3 The Special Functions Of Elementary Calculus
8.3.1 The Functions, sin,cos,exp
8.3.2 ln and log b
8.4 The Binomial Theorem
8.5 Exercises
8.6 L’Hôpital’s Rule
8.6.1 Interest Compounded Continuously
8.7 Exercises
8.8 Multiplication Of Power Series
8.9 Exercises
8.10 The Fundamental Theorem Of Algebra
8.11 Some Other Theorems

Chapter 9 Integration

9.1 Fundamental Definitions and Properties
9.2 The Fundamental Theorem of Calculus
9.3 Uniform Convergence And The Integral
9.4 A Simple Procedure For Finding Integrals
9.5 Stirling’s Formula
9.6 The Gamma Function
9.7 Laplace Transforms
9.8 Exercises

Chapter 10 Integration On Rough Paths∗

10.1 Finite p Variation
10.2 Piecewise Linear Approximation
10.3 The Young Integral

Chapter 11 Fourier Series

11.1 The Complex Exponential
11.2 Definition And Basic Properties
11.3 The Riemann Lebesgue Lemma
11.4 Dini’s Criterion For Convergence
11.5 Integrating And Differentiating Fourier Series
11.6 Ways Of Approximating Functions
11.6.1 Uniform Approximation With Trig. Polynomials
11.6.2 Mean Square Approximation
11.7 Exercises

Chapter 12 The Generalized Riemann Integral

12.1 Definitions And Basic Properties
12.2 Integrals Of Derivatives
12.3 Exercises
B.1 Closed and Open Sets
B.2 Compactness
B.2.1 Continuous Functions
B.2.2 Convergent Sequences
B.2.3 Continuity and the Limit of a Sequence
B.2.4 The Extreme Value Theorem and Uniform Continuity
B.2.5 Equivalence Of Norms
B.3 Norms On Linear Maps
B.4 Connected Sets
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