The Green Book of Mathematical Problems

Kenneth Hardy, Kenneth S. Williams

• Publisher: Dover Publications
• Year: 2013
• Print ISBN:
  9780486695730
• eBook ISBN:
  9780486169453
• Topic: Mathematics
• Subtopic: Linear Algebra
• Index: Mathematics

Rich selection of 100 practice problems — with hints and solutions — for students preparing for the William Lowell Putnam and other undergraduate-level mathematical competitions. Features real numbers, differential equations, integrals, polynomials, sets, other topics. Hours of stimulating challenge for math buffs at varying degrees of proficiency. References.

THE SOLUTIONS

It seemed that the next minute

they would discover a solution.

Yet it was clear to both of them

that the end was still far, far off,

and that the hardest and most complicated

part was only just beginning.

Anton Chekhov

1. If {b_n: n = 0, 1, 2, . . .} is a sequence of non-negative real numbers, prove that the series

converges for every positive real number a.

Solution: For a > 0 we set

so that

As b_n ≥ 0 we have, for n ≥ 1,

Now, for n ≥ 1, we may deduce that

Hence, for m ≥ 1, we have

As the partial sums s_m of (1.0) are bounded, the infinite series (1.0) converges for every a > 0.

2. Let a,b,c,d be positive real numbers, and let

Evaluate the limit

.

Solution: Considering the five cases specified in THE HINTS, we show that

We set

, so that c < kd.

(a) When b > d, as c + jd < (j + k)d, we have for n ≥ k

which tends to infinity as n tends to infinity, showing that L = + ∞,

(b) When b = d and a > c we have

which tends to infinity as n tends to infinity since the h...

Table of contents

• Cover
• Title Page
• Copyright Page
• Preface
• Dedication
• Contents
• Notation
• The Problems
• The Hints
• The Solutions
• Abbreviations
• References