query: Let $f:\mathbb{N}\to \mathbb{N}$ be bijection such that $\sum_{n=1}^{\infty} \frac{f(n)}{n^2}< \infty$
positive_passages.docid: 72804
positive_passages.text: Infinite series, injective function and rearrangement inequality
negative_passages.docid: 13979
negative_passages.text: Show that $\sum\limits_{n=0}^\infty \frac{z}{(1+z^2)^n}$ converges for all $z\in \mathbb{C}$ outside $|1+z^2|=1$
negative_passages.docid: 14975
negative_passages.text: Functions satisfying $f:\mathbb{N}\rightarrow\ \mathbb{N}$ and $f(f(n))+f(n+1)=n+2$
negative_passages.docid: 146436
negative_passages.text: Cardinality of the set of bijective functions on $\mathbb{N}$?
negative_passages.docid: 24107
negative_passages.text: Bijection $f:\mathcal{P}(A)\to(A\to \{0,1\})$
negative_passages.docid: 28461
negative_passages.text: Show that $\lim_{n\to\infty} \sum_{k=1}^{n} \frac{n}{n^2+k^2}=\frac{\pi}{4}$
negative_passages.docid: 11986
negative_passages.text: If $\sum a_n b_n <\infty$ for all $(b_n)\in \ell^2$ then $(a_n) \in \ell^2$
negative_passages.docid: 71162
negative_passages.text: Showing that $\sum_{n=1}^{\infty} \frac{z^n}{1+z^{2n}}$ converges
negative_passages.docid: 53105
negative_passages.text: How to prove that this sequence converges? $\sum_{n=1}^{\infty} \frac{1}{n\ln^2(n)}$
negative_passages.docid: 16143
negative_passages.text: How can I prove that $\sum_{n=1}^\infty \frac{1}{n(n+1)} = 1$?
negative_passages.docid: 67451
negative_passages.text: Building a bijection from a finite union of countable sets to $\mathbb{N}$
negative_passages.docid: 48251
negative_passages.text: Prove that if $n \in \mathbb{N}$, then $\sum_{d|n}{(d(n))^3}=(\sum_{d|n}{d(n)})^2$ where $d(n)$ is the divisor function.
negative_passages.docid: 82675
negative_passages.text: What is an example of function $f: \Bbb{N} \to \Bbb{Z}$ that is a bijection?
negative_passages.docid: 67271
negative_passages.text: $\sum_{n=-\infty}^{\infty}\frac{1}{(u +n)^2}=\frac{\pi^2}{(\sin \pi u)^2}$
negative_passages.docid: 71240
negative_passages.text: The limit of a sum $\sum_{k=1}^n \frac{n}{n^2+k^2}$ as $n\rightarrow\infty$
negative_passages.docid: 115411
negative_passages.text: if $A_n \longrightarrow \infty $ and $B_n \longrightarrow \infty $ then $(A_n+B_n) \longrightarrow \infty$
negative_passages.docid: 16776
negative_passages.text: How to prove $\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{m^2+n^2}=+\infty$