1) Show that $f_k(z)=z^k/k$ converges uniformly for $|z|<1$ 2) Show that $f_k'(z)=z^{k-1}$ does not converge uniformly for $|z|<1$. My Try: I did part 1. In part 2, I can prove that $f_k'(z)=z^{k-1}$ does not converge uniformly for $|z|\leq 1$. But how can I prove it for $|z|<1$? Can anybody please help me to figure it […]

I’m trying to find out if $\sum_{n=0}^\infty e^{-|x-n|}$ converges uniformly in $(\infty,\infty)$. Here’s my attempt at an answer: If $ x\in(-\infty,0]:$ $$e^{-|x-n|}=e^{(-(x-n))}=e^{x-n}\le e^{-n}$$ $\sum e^{-n}$ converges, therefore $\sum_{n=0}^\infty e^{-|x-n|}$ converges uniformly in $(-\infty,0]$. If $x\in[0,\infty):$ Here no series $\sum a_n$ exists so that $e^{-|x-n|} \le a_n$ for every $x\in[0,\infty)$, because for every n, there exists […]

I wonder if this is true: Let $(f_n)$ be a sequence of real-valued functions defined on a set $S\subset\mathbb{R}$. Assume that the function $f(x)$ is continuous and $\lim_{n\to\infty}f_n(x)=f(x)$. Does this imply that $f_n(x)$ is uniform convergent? The other way around it seems to be true, according to my book: The uniform limit of continuous function […]

I need to show that if $f$ is continuous function ($f:\mathbb{R}\rightarrow \mathbb{R}$), then there exists a sequence of polynomials which converges to $f$ on any compact subset of $\mathbb{R}$. I would appreciate any help/direction …

I want to show that, for $\alpha = \frac{1}{2}$, $$\lim_{x \to 0} \sum_{n=1}^{\infty}\frac{x}{(1+nx^2)n^{\alpha}}>0$$ Any ideas are welcome. (In a previous question, I considered the case for $\alpha > \frac{1}{2}$, from which we were able to derive a uniform upper bound and use Weierstrass M-test and Dominated Convergence Theorem to establish the continuity of the series […]

From an exercise list: Prove that if a sequence of continuous functions $f_n:X\rightarrow \mathbb{R}$ is such that $x_n\in X$, $\lim x_n = x \in X \Rightarrow \lim f_n(x_n) = f(x)$, then $f_n\rightarrow f$ uniformly on every compact subset of $X$. This question is very similar to this and also this one. But in this case, […]

We’ve been going over Uniform convergence in my class, but I’m very uncertain as to how to effectively show the correct proof. The $\lim_{n \rightarrow \infty}f(x) = 1$, understood using L’Hopital indeterminate form, which is not rigorously shown. (Help needed) I’m having the most trouble showing that for $a > 0, \{f_n\}$ converges uniformly to […]

Let $F_n, \ F$ be distribution functions with respect to some variables $X_n,\ X$ (in a not necessarily common probability space). Suppose that $F$ is continuous and $F_n \overset{d}{\rightarrow}F$ (i.e in law). Prove that $(F_n)$ converges uniformly to $F$, i.e. $$\displaystyle \lim_{n\rightarrow +\infty}\sup_{x\in \mathbb{R}}|F_n(x)-F(x)|=0$$ Comments. On a proof by contadiction, I ‘d suppose that for […]

Let $(S,d)$ be a complete separable metric space and consider the set $L^1(S,\mathbb{R})$ of functions $f:S \rightarrow \mathbb{R}$ which are 1-Lipschitz, i.e. $\forall x,y \in S: |f(x) – f(y)| \leq d(x,y)$. Further let: $$ \mathcal{P}^1(S) := \{P: \mathcal{B}_S \rightarrow [0,1] \mid P \mbox{ probability measure:} \forall a\in S: \int d(a,x) P(dx) < \infty \}, $$ […]

Consider a sequence of stochastic processes, $X_n(x)$ and a limiting process $X(x)$. For a fixed $x$, if $\mathbb{P}(X_n(x) \leq y)$ converges to $\mathbb{P}(X(x) \leq y)$ for continuity points of $F_{X(x)}(y) = \mathbb{P}(X(x) \leq y) $, then $\mathbb{E}[f(X_n(x)]$ converges to $\mathbb{E}[f(X(x)]$ for any $f$ which is continuous and bounded. Suppose I have uniform convergence in distribution […]

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