1. Definition (Sequences of Functions)
Let { f n } n ∈ N f D ⊂ R ( f n ) n converges uniformly on D f ϵ > 0 n 0 ∈ N depending only on ϵ ) such that:
You can't use 'macro parameter character #' in math mode |f_n(x) - f(x)| < \epsilon $$ for all $n > n_0$ and for **every** $x \in D$. ## 2. Characterization A sequence of functions $(f_n)_n$ converges uniformly to $f$ on $D$ if and only if: |f_n(x) - f(x)| < \epsilon $$ for all $n > n_0$ and for **every** $x \in D$. ## 2. Characterization A sequence of functions $(f_n)_n$ converges uniformly to $f$ on $D$ if and only if: \sup_{x \in D} |f_n(x) - f(x)| \to 0 \text{ as } n \to \infty
You can't use 'macro parameter character #' in math mode This means the maximum (or supremum) distance between $f_n$ and $f$ over the entire domain vanishes as $n$ increases. ## 3. Cauchy Criterion for Uniform Convergence A sequence $(f_n)_n$ converges uniformly on $D$ if and only if for every $\epsilon > 0$ there exists $n_0 \in \mathbb{N}$ such that for all $p, q > n_0$ (with $p \le q$): $$|f_p(x) - f_q(x)| < \epsilon \quad \forall x \in D$$. ## 4. Uniform Convergence of Series A series of functions $\sum_{n=1}^{+\infty} f_n$ converges uniformly on $D$ if the sequence of its **partial sums** $S_n(x) = \sum_{k=1}^{n} f_k(x)$ converges uniformly on $D$. ### Weierstrass M-Test A series $\sum f_n$ converges uniformly on $D$ if there exists a sequence of positive real numbers $M_n$ such that: 1. $|f_n(x)| \le M_n$ for all $x \in D$ and $n \in \mathbb{N}$. 2. The numerical series $\sum M_n$ is convergent. ## 5. Key Properties - **Continuity:** If $(f_n)_n$ is a sequence of continuous functions that converges uniformly to $f$, then $f$ is also continuous on $D$. - **Integration:** If $(f_n)_n$ converges uniformly to $f$ on $[a, b]$, and the functions are R-integrable, then the limit of the integrals is equal to the integral of the limit: $$\lim_{n \to \infty} \int_a^b f_n(x) , dx = \int_a^b f(x) , dx$$. --- **Note:** Uniform convergence is a stronger condition than pointwise convergence, where $n_0$ may depend on both $\epsilon$ and the specific point $x$. This means the maximum (or supremum) distance between $f_n$ and $f$ over the entire domain vanishes as $n$ increases. ## 3. Cauchy Criterion for Uniform Convergence A sequence $(f_n)_n$ converges uniformly on $D$ if and only if for every $\epsilon > 0$ there exists $n_0 \in \mathbb{N}$ such that for all $p, q > n_0$ (with $p \le q$): $$|f_p(x) - f_q(x)| < \epsilon \quad \forall x \in D$$. ## 4. Uniform Convergence of Series A series of functions $\sum_{n=1}^{+\infty} f_n$ converges uniformly on $D$ if the sequence of its **partial sums** $S_n(x) = \sum_{k=1}^{n} f_k(x)$ converges uniformly on $D$. ### Weierstrass M-Test A series $\sum f_n$ converges uniformly on $D$ if there exists a sequence of positive real numbers $M_n$ such that: 1. $|f_n(x)| \le M_n$ for all $x \in D$ and $n \in \mathbb{N}$. 2. The numerical series $\sum M_n$ is convergent. ## 5. Key Properties - **Continuity:** If $(f_n)_n$ is a sequence of continuous functions that converges uniformly to $f$, then $f$ is also continuous on $D$. - **Integration:** If $(f_n)_n$ converges uniformly to $f$ on $[a, b]$, and the functions are R-integrable, then the limit of the integrals is equal to the integral of the limit: $$\lim_{n \to \infty} \int_a^b f_n(x) , dx = \int_a^b f(x) , dx$$. --- **Note:** Uniform convergence is a stronger condition than pointwise convergence, where $n_0$ may depend on both $\epsilon$ and the specific point $x$.