query: The limit superior of $|\sum_{i=1}^n x_i^k|^{1/k}$ where $x_i\in\mathbb{C}$
positive_passages.docid: 16718
positive_passages.text: Doubt about step of the proof that $\limsup\limits_{n\to\infty}\left|\sum\limits_{\ell=1}^mz^n_\ell\right|^{1/n}=\max\limits_{1\le\ell\le m}|z_\ell|$
negative_passages.docid: 27748
negative_passages.text: Convergence of $\sum \frac{a_n}{S_n ^{1 + \epsilon}}$ where $S_n = \sum_{i = 1} ^ n a_n$
negative_passages.docid: 4643
negative_passages.text: The limit of truncated sums of harmonic series, $\lim\limits_{k\to\infty}\sum_{n=k+1}^{2k}{\frac{1}{n}}$
negative_passages.docid: 51694
negative_passages.text: Radius of convergence of a power series with $a_n$ convergent
negative_passages.docid: 54801
negative_passages.text: Limit of some specific "almost Riemann" sums
negative_passages.docid: 111205
negative_passages.text: Proving that $\lim_{k\to \infty}\frac{f(x_k)-f(y_k)}{x_k-y_k}=f'(c)$ where $\lim_{k\to \infty}x_k=\lim_{k\to \infty}y_k=c$ with $x_k<c<y_k$
negative_passages.docid: 101017
negative_passages.text: What is the sum of $\sum\limits_{i=1}^{n}ip^i$?
negative_passages.docid: 104996
negative_passages.text: Limit of $\lim_{n\rightarrow \infty} \sum^{n}_{i=1} \frac{1}{\sqrt {n^2 +i} }$
negative_passages.docid: 15956
negative_passages.text: Divergence of $\sum\limits_{k=1}^n \frac{a_n}{S_n}$ where $S_n = \sum\limits_{k=1}^n a_k$, when $S_n$ diverges
negative_passages.docid: 966
negative_passages.text: Show that $\lim_{n\rightarrow \infty} \sqrt[n]{c_1^n+c_2^n+\ldots+c_m^n} = \max\{c_1,c_2,\ldots,c_m\}$
negative_passages.docid: 116811
negative_passages.text: Proofs with limit superior and limit inferior: $\liminf a_n \leq \limsup a_n$
negative_passages.docid: 31889
negative_passages.text: The limit as $n$ approaches infinity of $n(a^{1/n}-1)$
negative_passages.docid: 10211
negative_passages.text: Proving that $\sum\limits_{k=0}^{n} {{m+k} \choose{m}} = { m+n+1 \choose m+1 }$
negative_passages.docid: 53688
negative_passages.text: How can one prove that $\lim_{n \to \infty}a^{1/n}=1$ for every $a>0$?
negative_passages.docid: 51107
negative_passages.text: Limit distribution of $\frac1n\cdot \max(X_1,X_2,...,X_n)$ for $(X_n)$ i.i.d. standard Cauchy
negative_passages.docid: 51555
negative_passages.text: Strictly increasing sequence $\{ a_k \}$ of positive integers such that $\sum 1/{a_k}$ is finite
negative_passages.docid: 103124
negative_passages.text: Proving that $f(z)=\sum\limits_{n=1}^{\infty}\frac{z}{n(1+n|z|^{2})}$ converges uniformly on $\mathbb{C}$