Monthly Problem Gems 1st Edition by Hongwei Chen ISBN 9780367766788 0367766787

1 Limits

1.1 A rational recurrence

1.2 Asymptotic behavior for a Pólya type recurrence

1.3 A nonlinear recurrence with Fibonacci exponents

1.4 Rate of convergence for an integral

1.5 How closely does this sum approximate the integral?

1.6 An arctangent series

1.7 Geometric mean rates

1.8 Limits of the weighted mean ratio

1.9 Limits of mean recurrences

1.10 A disguised half-angle iteration

1.11 Nested radicals and generalized Fibonacci numbers

1.12 A limit involving arctangent

1.13 Summing to the double factorials

1.14 A fractional part sum with Euler’s constant

1.15 A Putnam/Monthly limit problem

2 Infinite Series

2.1 Wilf wants us thinking rationally

2.2 Old wine in a new bottle

2.3 Another Euler sum

2.4 Reciprocal Catalan number sums

2.5 Catalan generating function

2.6 A series with log and harmonic numbers

2.7 A nonlinear harmonic sum

2.8 A series involving Riemann zeta values

2.9 Abel theorem continued

2.10 A convergence test

2.11 A power series with an exponential tail

2.12 An infinite matrix product

3 Integrations

3.1 A Lobachevsky integral

3.2 Two log gamma integrals

3.3 Short gamma products with simple values

3.4 Evaluate an integral by Feynman’s way

3.5 Three ways to evaluate a log-sine integral

3.6 A log-product integral

3.7 A Putnam problem beyond

3.8 A surface integral with many faces

3.9 Evaluate a definite integral by the gamma function

3.10 Digamma via a double integral

3.11 Another double integral

3.12 An integral with log and arctangent

3.13 Another integral with log and arctangent

3.14 An orthonormal function sequence

3.15 An definite integral Quickie

4 Inequalities

4.1 An inequality from Klamkin

4.2 Knuth’s exponential inequality

4.3 Tight bounds for the normal distribution

4.4 An inequality due to Knopp

4.5 A discrete inequality by integral

4.6 An inequality by power series

4.7 An inequality by differential equation

4.8 Bounds for a reciprocal of log sum

4.9 Log-concavity of a partial binomial sum

4.10 A bound of divisor sums related to the Riemann hypothesis

5 Monthly Miniatures

5.1 Value defined by an integral

5.2 Another mean value theorem

5.3 The product of derivatives by Darboux’s theorem

5.4 An integral-derivative inequality

5.5 Eigenvalues of a (0,1)-matrix

5.6 A matrix of secants

5.7 A gcd-weighted trigonometric sum

5.8 A lcm-sum weighted Dirichlet series

5.9 An infinite sum-product identity

5.10 A polynomial zero identity

A List of Problems

B Glossary

Bibliography

Index