query: Compact convergence $\sum_{n = 1}^{\infty} f(z^n)$
positive_passages.docid: 147764
positive_passages.text: show that $f(z)+f(z^2)+\cdots + f(z^n)+\cdots$ converges locally uniformly to an analytic function in the unit disk.
negative_passages.docid: 112510
negative_passages.text: Prove a.s. convergence of $(X_n)_n$ satisfying $E(X_{n+1} \mid F_n) \leq X_n+Y_n$ for $\sum_n Y_n<\infty$
negative_passages.docid: 149055
negative_passages.text: Uniform convergence of $\sum\limits_{k=1}^{\infty}\dfrac{kf(x^2)}{1+k^3f(x^2)^2}$
negative_passages.docid: 135110
negative_passages.text: Pointwise convergence of $h_n(x) = x^{1+\frac{1}{2n-1}}$
negative_passages.docid: 141218
negative_passages.text: Does $\sum_{n=1}^{\infty} \frac{\sin(n)}{n}$ converge conditionally?
negative_passages.docid: 9921
negative_passages.text: Convergence of $\sum_{n=1}^\infty\frac{n}{(n+1)!}$
negative_passages.docid: 14050
negative_passages.text: Determining precisely where $\sum_{n=1}^\infty\frac{z^n}{n}$ converges?
negative_passages.docid: 16480
negative_passages.text: Does $\sum_{n=1}^{\infty}\ln(n\sin(\frac{1}{n}))$ converge?
negative_passages.docid: 153063
negative_passages.text: Uniform convergence of $f_n(x)=nx^n(1-x)$ for $x \in [0,1]$?
negative_passages.docid: 41868
negative_passages.text: Radius of Convergence - $\sum_{n=1}^{\infty}2^n x^{n^2}$
negative_passages.docid: 146756
negative_passages.text: Convergence at infinitely many points in a compact set and convergence in the whole region
negative_passages.docid: 102581
negative_passages.text: Does $\sum_{n=1}^{\infty} \frac{1}{n^{1 + 1/n}}$ converge?
negative_passages.docid: 118105
negative_passages.text: What is the radius of convergence of $\sum z^{n!}$?
negative_passages.docid: 60891
negative_passages.text: Does $\sum\limits_{k=1}^\infty |\sin(ak)/k|$ converge?
negative_passages.docid: 22450
negative_passages.text: Convergence of $\sum\limits_{n=2}^\infty \frac{1}{n^\alpha \ln^\beta (n)} $ for nonnegative $\alpha$ and $\beta$
negative_passages.docid: 76593
negative_passages.text: Show that $f_n(z)=z^n$ converges locally uniformly on $|z|<1$ but not uniformly(?)
negative_passages.docid: 38752
negative_passages.text: Uniform convergence of $\sum (-1)^nf_n(x)$ on $[0,1]$ where $f_n(x)=x^n(1-x)$.