Infi-2 12

Function sequences #definition

\begin{matrix} Function sequence is a sequence in which each element is a function \\ For example: \\ f_{n} (x) = x^{n} \\ f_{1} (x) = x, f_{2} (x) = x^{2}, f_{3} (x) = x^{3}, \dots \end{matrix}

\begin{matrix} {f_{n} (x)} is called point-convergent on set A if \\ \forall x_{0} \in A : \forall ε > 0 : \exists N_{ε} : \forall n > N_{ε} : | f_{n} (x_{0}) - f (x_{0}) | < ε \\ f (x) is then called a pointwise limit of f_{n} \\ f_{n} \to f \\ Note: N_{ε} depends both on ε and x_{0} \in A \\ Hence pointwise convergence \end{matrix}

\begin{matrix} f_{n} (x) = x^{n} on [0, 1] \\ x < 1 ⟹ f_{n} (x) = x^{n} \to 0 \\ x = 1 ⟹ f_{n} (x) = 1^{n} = 1 \to 1 \\ f (x) = {\begin{matrix} 0 & x \in [0, 1) \\ 1 & x = 1 \end{matrix} \\ f_{n} \to f \end{matrix}

\begin{matrix} f_{n} (x) = x^{2} + \frac{x}{n} + \frac{7}{n^{2}} \\ lim_{n \to \infty} f_{n} (x) = lim_{n \to \infty} (x^{2} + \underset{\to 0}{\underset{⏟}{\frac{x}{n}}} + \underset{\to 0}{\underset{⏟}{\frac{7}{n^{2}}}}) = x^{2} \\ f (x) = x^{2} \\ f_{n} \to f \end{matrix}

\begin{matrix} lim_{n \to \infty} \frac{n^{2} x^{6}}{n^{2} + x^{6}} = lim_{n \to \infty} \frac{x^{6}}{1 + \underset{\to 0}{\underset{⏟}{\frac{x^{6}}{n^{2}}}}} = x^{6} \end{matrix}

\begin{matrix} lim_{n \to \infty} \sqrt{n^{2} x^{2} + x^{4}} - n x = lim_{n \to \infty} \frac{x^{4}}{\sqrt{n^{2} x^{2} + x^{4}} + n x} \\ x > 0 ⟹ f_{n} \to 0 \\ x = 0 ⟹ f_{n} (x) = 0 \\ x < 0 ⟹ f_{n} (x) = \underset{\to \infty}{\underset{⏟}{\sqrt{n^{2} x^{2} + x^{4}}}} - \underset{\to - \infty}{\underset{⏟}{n x}} \to \infty \\ f (x) = 0 on [0, \infty) \\ f_{n} \to f \end{matrix}

\begin{matrix} f_{n} (x) = n \arctan (\frac{x}{n}) \\ lim_{n \to \infty} \frac{\arctan (\frac{x}{n})}{\frac{1}{n}} \overset{t = \frac{1}{n}}{=} lim_{t \to 0} \frac{\arctan (t x)}{t} \overset{L}{=} lim_{t \to 0} \frac{x}{1 + (t x)^{2}} = x \end{matrix}

\begin{matrix} f_{n} (x) = n^{2} \ln (1 + \sin (\frac{x^{9}}{n^{2}})) \\ lim_{n \to \infty} n^{2} \ln (1 + \sin (\frac{x^{9}}{n^{2}})) = lim_{n \to \infty} \frac{x^{9} \cdot \ln (1 + \sin (\frac{x^{9}}{n^{2}}))}{\frac{\frac{x^{9}}{n^{2}}}{\sin (\frac{x^{9}}{n^{2}})} \cdot \sin (\frac{x^{9}}{n^{2}})} = \\ = lim_{t \to 0} x^{9} \cdot \frac{\ln (1 + \sin (t))}{\sin (t)} \cdot \frac{\sin (t)}{t} = x^{9} \\ x = 0 ⟹ f_{n} (x) = 0 = x^{9} \end{matrix}

\begin{matrix} f_{n} (x) = \sin^{4 n} (x) \\ lim_{n \to \infty} \sin^{4 n} (x) = lim_{n \to \infty} (\sin^{4} (x))^{n} = {\begin{matrix} 1 & \sin^{4} (x) = 1 \\ 0 & \sin^{4} (x) \neq 1 \end{matrix} = {\begin{matrix} 1 & x = \frac{π}{2} + π k \\ 0 & otherwise \end{matrix} \end{matrix}

\begin{matrix} f_{n} (x) = {\begin{matrix} 1 & x \in [0, \frac{1}{n}] \\ 0 & otherwise \end{matrix} on [0, 1] \\ x = 0 ⟹ f_{n} (x) = 1 \to 1 \\ x > 0 ⟹ \forall n > \frac{1}{x} : f_{n} (x) = 0 \to 0 \\ ⟹ f = {\begin{matrix} 1 & x = 0 \\ 0 & otherwise \end{matrix} on [0, 1] \end{matrix}

\begin{matrix} Let f_{n} \to f \\ Let f_{n} be continuous \\ Is f necessarily continuous? \\ No: f_{n} (x) = x^{n}, f (x) = {\begin{matrix} 1 & x = 1 \\ 0 & x \in [0, 1) \end{matrix} \\ Let f_{n}, f be differentiable \\ Does necessarily f_{n}^{'} \to f^{'} ? \\ No: f_{n} (x) = \frac{\sin (n^{8} x)}{n} \to 0 \\ f_{n}^{'} (x) = n^{7} \cos (n^{8} x) doesn’t converge \\ Let f_{n}, f be Riemann-integrable on [a, b] \\ Does necessarily \int_{a}^{b} f_{n} (x) d x \to \int_{a}^{b} f (x) d x ? \\ No: f_{n} (x) = {\begin{matrix} n & x \in [0, \frac{1}{n}] \\ 0 & otherwise \end{matrix} \to 0 \\ \int_{0}^{1} f_{n} (t) d t = \int_{0}^{1 / n} f_{n} (t) d t + \int_{\frac{1}{n}}^{1} f_{n} (t) d t = n t |_{t = 0}^{t = 1 / n} + C |_{t = \frac{1}{n}}^{t = 1} = 1 \end{matrix}