Prove that the function \(\cos \sqrt {x}\) is not periodic.
Old calculator I.
a) Suppose that we want to find \(\sqrt[3]{x}\) (\(x> 0\)) on a calculator that can find \(\sqrt{x}\) in addition to four ordinary arithmetic operations. Consider the following algorithm. A sequence of numbers \(\{y_n\}\) is constructed, in which \(y_0\) is an arbitrary positive number, for example, \(y_0 = \sqrt{\sqrt{x}}\), and the remaining elements are defined by \(y_{n + 1} = \sqrt{\sqrt{x y_n}}\) (\(n \geq 0\)).
Prove that \(\lim\limits_{n\to\infty} y_n = \sqrt[3]{x}\).
b) Construct a similar algorithm to calculate the fifth root.
The sequence of numbers \(a_1, a_2, a_3, \dots\) is given by the following conditions \(a_1 = 1\), \(a_{n + 1} = a_n + \frac {1} {a_n^2}\) (\(n \geq 0\)).
Prove that
a) this sequence is unbounded;
b) \(a_{9000} > 30\);
c) find the limit \(\lim \limits_ {n \to \infty} \frac {a_n} {\sqrt [3] n}\).
We are given rational positive numbers \(p, q\) where \(1/p + 1/q = 1\). Prove that for positive \(a\) and \(b\), the following inequality holds: \(ab \leq \frac{a^p}{p} + \frac{b^q}{q}\).
Let \(p\) and \(q\) be positive numbers where \(1 / p + 1 / q = 1\). Prove that \[a_1b_1 + a_2b_2 + \dots + a_nb_n \leq (a_1^p + \dots a_n^p)^{1/p}(b_1^q +\dots + b_n^q)^{1/q}\] The values of the variables are considered positive.
Draw all of the stairs made from four bricks in descending order, starting with the steepest \((4, 0, 0, 0)\) and ending with the shallowest \((1, 1, 1, 1)\).
Liouville’s discrete theorem. Let \(f (x, y)\) be a bounded harmonic function (see the definition in problem number 11.28), that is, there exists a positive constant \(M\) such that \(\forall (x, y) \in \mathbb {Z}^2\) \(| f (x, y) | \leq M\). Prove that the function \(f (x, y)\) is equal to a constant.
A frog jumps over the vertices of the triangle \(ABC\), moving each time to one of the neighbouring vertices.
How many ways can it get from \(A\) to \(A\) in \(n\) jumps?
Let \((1 + \sqrt {2} + \sqrt {3})^n = p_n + q_n \sqrt {2} + r_n \sqrt {3} + s_n \sqrt {6}\) for \(n \geq 0\). Find:
a) \(\lim \limits_ {n \to \infty} {\frac {p_n} {q_n}}\); b) \(\lim \limits_ {n \to \infty} {\frac {p_n} {r_n}}\); c) \(\lim \limits_ {n \to \infty} {\frac {p_n} {s_n}}\);
Find the generating functions of the sequences of Chebyshev polynomials of the first and second kind: \[F_T(x,z) = \sum_{n=0}^{\infty}T_n(x)z^n;\quad F_U(x,z) = \sum_{n=0}^{\infty}U_n(X)z^n.\]
Definitions of Chebyshev polynomials can be found in the handbook.