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# Excursions in Classical Analysis

Hongwei Chen

## Table of Contents

Preface

1 Two Classical Inequalities
1.1 AM-GM Inequality
1.2 Cauchy-Schwarz Inequality
Exercises
References

2 A New Approach for Proving Inequalities
Exercises
References

3 Means Generated by an Integral
Exercises
References

4 The L’Hôpital Monotone Rule
Exercises
References

5 Trigonometric Identities via Complex Numbers
5.1 A Primer of complex numbers
5.2 Finite Product Identities
5.3 Finite Summation Identities
5.4 Euler’s Infinite Product
5.5 Sums of inverse tangents
5.6 Two Applications
Exercises
References

6 Special Numbers
6.1 Generating Functions
6.2 Fibonacci Numbers
6.3 Harmonic numbers
6.4 Bernoulli Numbers
Exercises
References

7 On a Sum of Cosecants
7.1 A well-known sum and its generalization
7.2 Rough estimates
7.3 Tying up the loose bounds
7.4 Final Remarks
Exercises
References

8 The Gamma Products in Simple Closed Forms
Exercises
References

9 On the Telescoping Sums
9.1 The sum of products of arithmetic sequences
9.2 The sum of products of reciprocals of arithmetic sequences
9.3 Trigonometric sums
9.4 Some more telescoping sums
Exercises
References

10 Summation of Subseries in Closed Form
Exercises
References

11 Generating Functions for Powers of Fibonacci Numbers
Exercises
References

12 Identities for the Fibonacci Powers
Exercises
References

13 Bernoulli Numbers via Determinants
Exercises
References

14 On Some Finite Trigonometric Power Sums
14.1 Sums involving sec2p(kπ/n)
14.2 Sums involving csc2p(kπ/n)
14.3 Sums involving tan2p(kπ/n)
14.4 Sums involving cot2p(kπ/n)
Exercises
References

15 Power Series of (arcsin x)2
15.1 First Proof of the Series (15.1)
15.2 Second Proof of the Series (15.1)
Exercises
References

16 Six Ways to Sum ζ(2)
16.1 Euler’s Proof
16.2 Proof by Double Integrals
16.3 Proof by Trigonometric Identities
16.4 Proof by Power Series
16.5 Proof by Fourier Series
16.6 Proof by Complex Variables
Exercises
References

17 Evaluations of Some Variant Euler Sums
Exercises
References

18 Interesting Series Involving Binomial Coefficients
18.1 An integral representation and its applications
18.2 Some Extensions
18.3 Searching for new formulas for π
Exercises
References

19 Parametric Differentiation and Integration
Example 1
Example 2
Example 3
Example 4
Example 5
Example 6
Example 7
Example 8
Example 9
Example 10
Exercises
References

20 Four Ways to Evaluate the Poisson Integral
20.1 Using Riemann Sums
20.2 Using A Functional Equation
20.3 Using Parametric Differentiation
20.4 Using Infinite Series
Exercises
References

21 Some Irresistible Integrals
21.1 Monthly Problem 10611
21.2 Monthly Problem 11206
21.3 Monthly Problem 11275
21.4 Monthly Problem 11277
21.5 Monthly Problem 11322
21.6 Monthly Problem 11329
21.7 Monthly Problem 11331
21.8 Monthly Problem 11418

Exercises

References

Solutions to Selected Problems

Index