Mathematical progress depends on a stream of new problems, but most new problems emerge simply from well-established principles. For students who want to master existing principles and build their own knowledge of mathematics, the moral is simple: “Learn from the masters!” Often, historical developments of a particular topic provide the best way to learn that topic. In this chapter, by examining various proofs of the arithmetic mean and geometric mean (AM-GM) inequality and the Cauchy-Schwarz inequality, we see the motivation necessary to work through all the steps of a rigorous proof. Some of these proofs contain brilliant ideas and clever...

A beautiful approach for a specific problem frequently leads to wide applicability, thereby solving a host of new, related problems. We have seen in the preceding chapter some memorable approaches from a great variety of sources for proving the AM-GM inequality and the Cauchy-Schwarz inequality. There are many more different proofs recorded in the monograph [2] by Bullen et.al. In this regard, one may wonder whether there are any new approaches to re-prove these two inequalities? In this chapter, we introduce aunified elementary approachfor proving inequalities [3, 4]. We will see that this new approach not only recovers...

For a pair of distinct positive numbers,aandb, a number of different quantitiesM(a,b) are known asmeans:

1. the arithmetic mean:A(a,b) = (a+b)/2

2. the geometric mean:$G(a,b)=\sqrt{ab}$

3. the harmonic mean:H(a,b) = 2ab/(a+b)

4. the logarithmic mean:L(a,b) = (b−a)/(lnb− lna)

5. the Heronian mean:$N(a,b)=(a+\sqrt{ab}+b)/3$

6. the centroidal mean:T(a,b) = 2(a² +ab+b²)/3(a+b)

These are allpositively homogeneous, in the sense that\[M(\lambda a,\lambda b)=\lambda M(a,b)\quad \text{for}\ \text{all}\ \lambda >\text{0,}\]andsymmetric, in the sense thatM(a,b) =M(b,a). Moreover, all of the named means...

In elementary calculus, we learn that iff(x) is continuous and has a positive derivative on (a,b) thenf(x) is increasing on (a,b). However, if we are trying to prove the monotonicity of\[f(x)=\frac{{{x}^{p}}-1}{{{x}^{q}}-1},\quad (p>q>0,x\ge 1)\caption {(4.1)}\]in this way, we see the numerator of the derivative is\[(p-q){{x}^{p+q-1}}-p{{x}^{p-1}}+q{{x}^{q-1}},\]where the positivity is not immediately evident. In general, the derivative of a quotient is quite messy and the process to show monotonicity can be tedious. In this chapter, we introduce a general method for proving the monotonicity of a large class of quotients. Because of the similarity to the hypotheses to...

Trigonometric identities often arise in a variety of branches of mathematics, including geometry, analysis, number theory and applied mathematics. The classical standard tables [4] and [5] have collected many of these identities, such as\[\prod\limits_{k=1}^{n-1}{\sin \left( \frac{k\pi }{n} \right)=\frac{n}{{{2}^{n-1}}}.}\caption {(5.1)}\]

However, the proofs for these identities are widely scattered throughout books and journals. In this chapter, we present a comprehensive study on proving trigonometric identities via complex numbers. To keep the chapter to a reasonable page limit, we only examine several families of identities in [4, 5]. But, we emphasize that the methods used here can be applied to prove many identities. As applications of...

All numbers are not created equal. Some sequences of numbers in analysis seem quite complicated and mysterious, yet they have played an important role and have made many unexpected appearances in analysis and number theory. A large body of literature exists on these special numbers but much of it in widely scattered books and journals. In this chapter we investigate three of these special sets of numbers, the Fibonacci, harmonic and Bernoulli numbers. We derive some basic results that are readily accessible to those with a knowledge of elementary analysis. These materials will serve as a brief primer on these...

Undergraduate research has been a longstanding practice in the experimental sciences. Only recently, however, have significant numbers of undergraduates begun to participate in mathematical research. While many math students are interested in research, they often don’t know what problems are open or how to get started. Finding appropriate problems becomes of fundamental importance. Joseph Gallian has run an excellent undergraduate mathematics research program [4] at the University of Minnesota, Duluth since 1977. He offers the following criteria for successful undergraduate research problems:

not much background reading is required;

partial results are probable;

the problem has been posed recently;

new results...

To be a good problem solver, it is not just what you know, but how you use what you know. As we have seen in previous chapters, many formulas and theorems emerge from the analysis of specific calculations and special cases. Halmos once said “the source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly great generality is in essence the same as a small and concrete special case.” Thus, to solve a difficult problem, one should initially make the problem concrete and physical whenever...

In analysis, there are only a few series with partial sums that we can calculate in closed form and almost all of these come from telescoping sums. In this chapter, we apply this technique to a variety of problems, beginning with some that are arithmetic and trigonometric, moving to others that involve Apéry-like formulas and famous families of numbers, and finally ending with a class of problems that can be solved algorithmically.

We begin with Gauss’s legendary method for summing arithmetic series. Consider an arithmetic series ofnterms with first terma, last termband common differenced....

One of the problems in the Second Putnam Mathematical Competition (1939) was to prove that if\[u(x)=\sum\limits_{n=0}^{\infty }{\frac{{{x}^{3n}}}{(3n)!},\quad }v(x)=\sum\limits_{n=0}^{\infty }{\frac{{{x}^{3n+1}}}{(3n+1)!},\quad }w(x)=\sum\limits_{n=0}^{\infty }{\frac{{{x}^{3n+2}}}{(3n+2)!},}\]then${{u}^{3}}+{{v}^{3}}+{{w}^{3}}-3uvw=1$. The suggested solution relied on\[\begin{align*} u+v+w & = {{e}^{x}}, \\ u+\omega v+{{\omega }^{2}}w & = {{e}^{\omega x}}, \\ u+{{\omega }^{2}}v+\omega w & = {{e}^{{{\omega }^{2}}x}}, \\ \end{align*}\]where$\omega ={{e}^{2\pi i/3}}$, a primitive cubic root of unity. Notice that allu,vandware subseries of${{e}^{x}}$. As a byproduct, solving the above system of equations yields\[\begin{array}{ll} u(x)=\frac{1}{3}\left( {{e}^{x}}+2{{e}^{-x/2}}\cos \left( \frac{\sqrt{3}x}{2} \right) \right), \\ v(x)=\frac{1}{3}\left( {{e}^{x}}-{{e}^{-x/2}}\cos \left( \frac{\sqrt{3}x}{2} \right)+\sqrt{3}{{e}^{-x/2}}\sin \left( \frac{\sqrt{3}x}{2} \right) \right), \\ w(x)=\frac{1}{3}\left( {{e}^{x}}-{{e}^{-x/2}}\cos \left( \frac{\sqrt{3}x}{2} \right)-\sqrt{3}{{e}^{-x/2}}\sin \left( \frac{\sqrt{3}x}{2} \right) \right). \\ \end{array}\]

Later, for any positive integerkandm= 0, 1,…,k− 1, Rubel and Stolarsky in [12] proved\[\sum\limits_{n=0}{\frac{{{x}^{kn+m}}}{(kn+m)!}=\frac{1}{k}\left( {{e}^{x}}+\frac{{{e}^{\omega x}}}{{{\omega }^{m}}}+\frac{{{e}^{{{\omega }^{2}}x}}}{{{\omega }^{2m}}}+\cdots +\frac{{{e}^{{{\omega }^{k-1}}x}}}{{{\omega }^{(k-1)m}}} \right)},\caption {(10.1)}\]

where$\omega ={{e}^{2\pi i/k}}$is a primitivekth root of unity.

These results motivate a question that does not seem to have received much...

In this chapter, using Binet’s formula and the power-reduction formulas, we derive explicit formulas of generating functions for powers of Fibonacci numbers. The corresponding results are extended to the famous Lucas and Pell numbers.

As we did in Chapter 6, by${{F}_{n+2}}={{F}_{n+1}}+{{F}_{n}}$, we found that the generating function for Fibonacci numbers is\[\sum\limits_{n=1}^{\infty }{{{F}_{n}}{{x}^{n}}=\frac{x}{1-x-{{x}^{2}}}}.\caption {(11.1)}\]

In general, for any positive integerk, let the generating function of$\left\{ F_{n}^{k} \right\}$be\[{{G}_{k}}(x)=\sum\limits_{n=1}^{\infty }{F_{n}^{k}{{x}^{n}}.}\]

It is natural to ask whether there is an explicit formula for${{G}_{k}}(x)$. Whenk= 2, there is a recursive relation\[F_{n+3}^{2}=2F_{n+2}^{2}+2F_{n+1}^{2}-F_{n}^{2}.\caption {(11.2)}\]

To prove this, using Binet’s formula\[{{F}_{n}}=\frac{1}{\sqrt{5}}({{\alpha }^{n}}-{{\beta }^{n}}),\caption {(11.3)}\]...

In this chapter, using generating functions, we establish the following identities for any positive integers>nandk,\[F_{n+k+1}^{k}=\sum\limits_{i=0}^{k}{{{a}_{i}}(k)F_{n+k-i,}^{k}}\]where thea_i(k) are given explicitly in terms of Fibonomial coefficients (see (12.7) below). Along the way, we will focus on how to derive identities, instead of merely focusing on verification.

Recall that Fibonacci numbers {F_n} are defined by\[\begin{align*} {{F}_{1}} & = {{F}_{2}}=1,\\ {{F}_{n+2}} & = {{F}_{n+1}}+{{F}_{n}}\quad (\text{for }n\ge 1).\end{align*}\caption {(12.1)}\]

In the previous chapter, we proved that for anyn≥ 1,\[F_{n+3}^{2}=2F_{n+2}^{2}+2F_{n+1}^{2}-F_{n}^{2},\caption {(12.2)}\]\[F_{n+4}^{3}=3F_{n+3}^{3}+6F_{n+2}^{3}-3F_{n+1}^{3}-F_{n}^{3}.\caption {(12.3)}\]

In these identities, a power of a Fibonacci number is expressed as a linear combination of the same power of successive Fibonacci numbers. Naturally, we ask whether...

In this chapter, we represent Bernoulli numbers via determinants based on their recursive relations and Cramer’s rule. These enable us to evaluate a class of determinants involving factorials where the evaluation of these determinants by row and column manipulation is either quite challenging or almost impossible [3].

Recall the recursive relation for the Bernoulli numbersB_k(see (6.12)):\[{{B}_{0}}=1,\quad \sum\limits_{k=0}^{n-1}{\left( \begin{matrix} n \\ k \\ \end{matrix} \right){{B}_{k}}=0.}\caption {(13.1)}\]

Letb_k=B_k/k!. Rewriting (13.1) explicitly yields the following system of linear equations:\[\left\{ \begin{array}{ll} & {{b}_{1}}+\frac{1}{2!}=0, \\ & \frac{{{b}_{1}}}{2!}+{{b}_{2}}+\frac{1}{3!}=0, \\ & \frac{{{b}_{1}}}{3!}+\frac{{{b}_{2}}}{2!}+{{b}_{3}}+\frac{1}{4!}=0, \\ & \cdots \cdots \\ & \frac{{{b}_{1}}}{(n-1)!}+\frac{b2}{(n-2)!}+\cdots +{{b}_{n-1}}+\frac{1}{n!}=0, \\ & \frac{{{b}_{1}}}{n!}+\frac{{{b}_{2}}}{(n-1)!}+\cdots +{{b}_{n}}+\frac{1}{(n+1)!}=0, \\ \end{array} \right.\]or equivalently\[\left\{ \begin{align*} & {{b}_{1}}=-\frac{1}{2!}, \\ & \frac{{{b}_{1}}}{2!}+{{b}_{2}}=-\frac{1}{3!}, \\ & \frac{{{b}_{1}}}{3!}+\frac{{{b}_{2}}}{2!}+{{b}_{3}}=-\frac{1}{4!}, \\ & \cdots \cdots \\ & \frac{{{b}_{1}}}{(n-1)!}+\frac{b2}{(n-2)!}+\cdots +{{b}_{n-1}}=-\frac{1}{n!}, \\ & \frac{{{b}_{1}}}{n!}+\frac{{{b}_{2}}}{(n-1)!}+\cdots +{{b}_{n}}=-\frac{1}{(n+1)!}, \\ \end{align*} \right.\]

Solving forb_nby means of Cramer’s rule, we find\[{{b}_{n}}=\left| \begin{matrix} 1 & 0 & 0 & \cdots & -\frac{1}{2!} \\ \frac{1}{2!} & 1 & 0 & \cdots & -\frac{1}{3!} \\ \frac{1}{3!} & \frac{1}{2!} & 1 & \cdots & -\frac{1}{4!} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \frac{1}{(n-1)!} & \frac{1}{(n-2)!} & \frac{1}{(n-3)!} & \cdots & -\frac{1}{n!} \\ \frac{1}{n!} & \frac{1}{(n-1)!} & \frac{1}{(n-2)!} & \cdots & -\frac{1}{(n+1)!} \\ \end{matrix} \right|.\]

Using the basic properties of the determinants, factoring out the coefficient −1...

In Chapter 5, we established the trigonometric identity (see (5.22))\[\sum\limits_{k=0}^{n-1}{\frac{1}{1-2x\cos \left( \frac{2k\pi }{n} \right)+{{x}^{2}}}=\frac{n(1+{{x}^{n}})}{(1-{{x}^{n}})(1-{{x}^{2}})},}\]and reproduced the following elegant formula (see (5.2))\[\sum\limits_{k=1}^{n-1}{{{\cos }^{2}}\left( \frac{k\pi }{n} \right)}=\frac{(n-1)(n+1)}{3}\]along with\[\sum\limits_{k=1}^{n-1}{{{\tan }^{2}}\left( \frac{k\pi }{n} \right)}=n(n-1),\quad \text{if}\ n\ \text{is}\ \text{odd}.\]

In contrast to Fourier series, these finite power sums are over angles equally dividing the upper half plane. Moreover, these beautiful and somewhat surprising sums often arise in both analysis and number theory. In this chapter, by using generating functions, we extend the above results to the power sums as shown in identities (14.9)–(14.20) and in the Appendix.

We begin with establishing two auxiliary trigonometric identities. To this end, recall (5.15)\[\prod\limits_{k=1}^{n-1}{(\cos \theta -\cos (k\pi /n))={{2}^{1-n}}\frac{\sin n\theta }{\sin \theta }}.\caption {(14.1)}\]

Taking the logarithm of...

The power series expansion\[{{(\arcsin\ x)}^{2}}=\frac{1}{2}\sum\limits_{n=1}^{\infty }{\frac{{{(2x)}^{2n}}}{{{n}^{2}}\left( \begin{matrix} 2n \\ n \\ \end{matrix} \right)},\quad |x|\le 1}\caption {(15.1)}\]plays an important role in the evaluation of series involving the central binomial coefficient. Some classical examples [1] are\[\sum\limits_{n=1}^{\infty }{\frac{1}{{{n}^{2}}\left( \begin{matrix} 2n \\ n \\ \end{matrix} \right)}}=\frac{{{\pi }^{2}}}{18}=\frac{1}{3}\zeta (2),\]\[\sum\limits_{n=1}^{\infty }{\frac{{{(-1)}^{n+1}}}{{{n}^{3}}\left( \begin{matrix} 2n \\ n \\ \end{matrix} \right)}}=\frac{2}{5}\zeta (3),\]\[\sum\limits_{n=1}^{\infty }{\frac{1}{{{n}^{4}}\left( \begin{matrix} 2n \\ n \\ \end{matrix} \right)}}=\frac{17}{3456}{{\pi }^{4}}=\frac{17}{36}\zeta (4),\]where ζ(x) is the Riemann zeta function. Since Euler, the series (15.1) has been established in a variety of ways, for example, by using Euler’s series transformation [2] or by using the Gregory series for arctanx[1]. There are also a number of elegant and clever elementary proofs. In this chapter, we present two such proofs and display some applications. The first proof is based on the power series solution to a...

The history of mathematics is valuable because it shows the motivations underlying the developments of many branches of mathematics. Euler’s recorded work on infinite series provides a particularly instructive example. In the eighteenth century infinite series were, as they are today, considered an essential part of analysis. In comparison, at the dawn of the seventeenth century infinite series were little understood and infrequently encountered. For example, Jakob Bernoulli handled the divergence of the harmonic series perfectly, but he could not find the exact value of the convergent series\[\zeta (2):=1+\frac{1}{{{2}^{2}}}+\frac{1}{{{3}^{2}}}+\cdots +\frac{1}{{{n}^{2}}}+\cdots .\]

Indeed, he included in hisTractatus[3] a plea for help...

In the mathematical world Euler is akin to the likes of Shakespeare and Mozart — universal, richly detailed, and inexhaustible. His work, dating back to the early eighteenth century, is still very much alive and generating intense interest.

In response to a letter from Goldbach, for integersm≥ 1 andn≥ 2, Euler considered sums of the form\[S(m,n):={{\sum\limits_{k=1}^{\infty }{\frac{1}{{{(k+1)}^{n}}}\left( 1+\frac{1}{2}+\cdots +\frac{1}{k} \right)}}^{m}}=\sum\limits_{k=1}^{\infty }{\frac{1}{{{(k+1)}^{n}}}H_{k}^{m},}\caption {(17.1)}\]and was successful in obtaining several explicit values of these sums in terms of the Riemann zeta function ζ(k). Indeed, he established the beautiful formula\[S(1,2)=\sum\limits_{k=1}^{\infty }{\frac{1}{{{(1+k)}^{2}}}{{H}_{k}}=\zeta }(3)\]as well as the more general relations: for alln> 2,\[S(1,n)=\frac{n}{2}\zeta (n+1)-\frac{1}{2}\sum\limits_{k=1}^{n-2}{\zeta (k+1)\zeta (n-k).}\caption {(17.2)}\]

Two...

In his wonderful paper [4], Lehmer defines a series to beinterestingif the sum has a closed form in terms of well-known constants. Using the power series\[\frac{2x\arcsin\ x}{\sqrt{1-{{x}^{2}}}}=\sum\limits_{n=1}^{\infty }{\frac{{{(2x)}^{2n}}}{n\left( \begin{matrix} 2n \\ n \\ \end{matrix} \right)},}\caption {(18.1)}\]he establishes a large class of interesting series, including\[\sum\limits_{n=1}^{\infty }{\frac{1}{\left( \begin{matrix} 2n \\ n \\ \end{matrix} \right)}=\frac{1}{2}+\frac{1}{6}+\frac{1}{20}+\frac{1}{70}+\cdots =\frac{1}{3}+\frac{2\sqrt{3}}{27}\pi ,}\caption {(18.2)}\]\[\sum\limits_{n=1}^{\infty }{\frac{1}{n\left( \begin{matrix} 2n \\ n \\ \end{matrix} \right)}=\frac{1}{2}+\frac{1}{12}+\frac{1}{60}+\frac{1}{280}+\cdots =\frac{\sqrt{3}}{9}\pi ,}\caption {(18.3)}\]and\[\sum\limits_{n=1}^{\infty }{\frac{1}{{{n}^{2}}\left( \begin{matrix} 2n \\ n \\ \end{matrix} \right)}}=\frac{1}{2}+\frac{1}{24}+\frac{1}{180}+\frac{1}{1120}+\cdots =\frac{1}{18}{{\pi }^{2}}.\caption {(18.4)}\]

However, he points out that the series\[\sum\limits_{n=1}^{\infty }{\frac{1}{{{n}^{3}}\left( \begin{matrix} 2n \\ n \\ \end{matrix} \right)}}=\frac{1}{2}+\frac{1}{48}+\frac{1}{540}+\frac{1}{4480}+\cdots =4\int_{0}^{1/2}{\frac{{{(\arcsin\ y)}^{2}}}{y}dy}\]is a “higher transcendent” and does not admit a simple closed form.

Motivated by such results, it is natural for us to ask when a given series involving the central binomial coefficients is interesting. In this chapter, we focus our attention on a class of series that possess integral representations....

In this chapter, we present an integration method that evaluates integrals via differentiation and integration with respect to a parameter. This approach has been a favorite tool of applied mathematicians and theoretical physicists. In his autobiography [3], eminent physicist Richard Feynman mentioned how he frequently used this approach when confronted with difficult integrations associated with mathematics and physics problems. He referred to this approach as “a different box of tools”. However, most modern texts either ignore this subject or provide only few examples. In the following, we illustrate this method with the help of some selected examples, most of them...

In general, it is difficult to decide whether or not a given function can be integrated via elementary methods. In light of this, it is quite surprising that the value of the Poisson integral\[I(x)=\int_{0}^{\pi }{\text{ln}}(1-2x\cos \theta +{{x}^{2}})d\theta \]can be determined precisely. Even more surprising is that we can do so for every value of the parameterx. In this chapter, using four different methods, we show that\[I(x)=\left\{ \begin{array}{ll} 0,\ & \text{if }|x| < 1; \\ 2\pi \ln|x|,\ & \text{if }|x| > 1. \\\end{array} \right.\]

Our integral is one of several known as the Poisson integral. All are related in some way to Poisson’s integral formula, which recovers an analytic function on the disk from its boundary values, a...

Over the years, we have seen a great many interesting integral problems and clever solutions published in the Monthly. Most of them contain mathematical ingenuities. In this chapter, we record eight of these intriguing integrals. They are so striking and so elegant that we can not resist including an account of them in this book. I still remember how exciting I found them on first encounter. In presenting the combination of approaches required to evaluate these integrals, I have tried to follow the most interesting route to the results and endeavored to highlight connections to other problems and to more...

Here the solutions do not include any Monthly and Putnam problems. For Monthly problems, please refer to the journal. The second pair of numbers indicates when and where the solution has been published. For Putnam problems and solutions before 2000, see

1.The William Lowell Putnam Mathematical Competition. Problems and Solutions: 1938–1964, by A. M. Gleason, R. E. Greenwood and L. M. Kelly.

2.The William Lowell Putnam Mathematical Competition. Problems and Solutions: 1965–1984, by G. L. Alexanderson, L. F. Klosinski and L. C. Larson.

3.The William Lowell Putnam Mathematical Competition 1985–2000: Problems, Solutions and Commentary, by K. S....