Cristinel Mortici

The Splitting Method and Double Sequences

We use here the splitting method to establish some useful convergence results. The splitting method is a useful tool to compute the limits of certain sequences of real numbers, whose general form is a sum s n of n terms, which do not behave in the same way. In fact, this method consists of decomposing the sum s n into two sums, which are analyzed separately, using different methods in general, adapted to the behavior of the terms composing them. We give first some examples of problems which use the splitting method and then some general results, with lots of practical applications.

Some Applications of the Hamilton-Cayley Theorem

We saw in a previous chapter that the characteristic polynomial of a matrix and its zeros, the eigenvalues of the matrix, can give precious information. The purpose of this chapter is to present a powerful theorem due to Hamilton and Cayley that gives an even stronger relation between a matrix and its characteristic polynomial: in a certain sense, the matrix is a “root” of its characteristic polynomial. After presenting a proof of this theorem, we investigate some interesting applications.

The Extreme Value Theorem

The extreme value theorem asserts that any continuous function defined on a compact interval with real values is bounded and it attains its extrema. Indeed, let us consider a continuous function f: [a, b] → ℝ. If we assume by contradiction that f is unbounded, then for each positive integer n, we can find an element x n in [a, b] such that f(x n ) > n. In this way, we define a bounded sequence (x n )n ≥ 1. The Bolzano-Weierstrass theorem implies (due to the compactness of [a, b]) the existence of a convergent subsequence (x k n )n ≥ 1. We have f(x k n ) > k n , for all positive integers n. In particular, the sequence (f(x k n ))n ≥ 1 is unbounded. This is a contradiction, because the sequence (f(x k n ))n ≥ 1 is convergent to f(l), where l is the limit of the sequence (x k n )n ≥ 1. Then suppose that M = supx ∈ [a, b]f(x) is the least upper bound (or supremum) of f (which is finite, as shown above). There exists a sequence (u n )n ≥ 1 ⊆ [a, b] such that limn → ∞f(u n ) = M. By Bolzano-Weierstrass theorem again, (u n )n ≥ 1 has a convergent subsequence; let us call (u k n )n ≥ 1 this subsequence, and let c ∈ [a, b] be its limit. We then have M = limn → ∞f(u n ) = limn → ∞f(u k n ) = f(c), and thus f attains its supremum M. In a similar manner, we show that f attains its infimum, thus finishing the proof of the theorem. □

Derivatives and Functions’ Variation

The derivatives of a differentiable function \(f: [a,b] \rightarrow \mathbb{R}\) give us basic information about the variation of the function. For instance, it is well-known that if f ′ ≥ 0, then the function f is increasing and if f ′ ≤ 0, then f is decreasing. Also, a function defined on an interval having the derivative equal to zero is in fact constant. All of these are consequences of some very useful theorems due to Fermat, Cauchy, and Lagrange. Fermat’s theorem states that the derivative of a function vanishes at each interior extremum point of f. The proof is not difficult: suppose that x0 is a local extremum, let us say a local minimum. Then f(x0 + h) − f(x0) ≥ 0 for all h in an open interval (−δ, δ). By dividing by h and passing to the limit when h approaches 0, we deduce that f ′ (x0) ≥ 0 (for h > 0) and f ′ (x0) ≤ 0 (for h < 0); thus f ′ (x0) = 0.

The Number e

The basic symbol of mathematical analysis is the well-known number e. It is introduced as the limit of the sequence \(e_{n} = \left (1 + \frac{1} {n}\right )^{n},\ n \geq 1.\)

Polynomial Functions Involving Determinants

Some interesting properties of determinants can be established by defining polynomial functions of the type f(X) = det(A + XB), where A, B are n × n matrices with complex entries. Under this hypothesis, f is a polynomial of degree ≤ n. The coefficient of X n is equal to detB and the constant term of f is equal to detA.

Mathematical (and Other) Bridges

Many people who read this book will probably be familiar with the following result (very folkloric, if we may say so).

Riemann and Darboux Sums

Let \(f: [a,b] \rightarrow \mathbb{R}\) be continuous and positive. By the subgraph of f, we mean the region from the xy-plane delimited by the x-axis, the lines x = a, x = b, and the curve y = f(x). More precisely, the subgraph is the set

The Intermediate Value Theorem

\displaystyle{\left \{(x,y) \in \mathbb{R}^{2}\;\vert \;a \leq x \leq b,\;0 \leq y \leq f(x)\right \}.}