Matching Tag: irrationality
Is ∑ n = 1 ∞ 1 / ( 2 n − 3 ) \sum_{n=1}^{∞}1/(2^n-3) ∑ n = 1 ∞ 1/ ( 2 n − 3 )
Let a 1 , a 2 , … a_1,a_2,\ldots a 1 , a 2 , … a n → ∞ a_n\to \infty a n → ∞ ∑ n τ ( n ) a 1 ⋯ a n \sum_{n} \frac{\tau(n)}{a_1\cdots a_n} ∑ n a 1 ⋯ a n τ ( n ) τ ( n ) \tau(n) τ ( n )
Let F 1 = F 2 = 1 F_1=F_2=1 F 1 = F 2 = 1 F n + 1 = F n + F n − 1 F_{n+1}=F_n+F_{n-1} F n + 1 = F n + F n − 1 n 1 < n 2 < ⋯ n_1<n_2<\cdots n 1 < n 2 < ⋯ n k + 1 / n k ≥ c > 1 n_{k+1}/n_k \geq c>1 n k + 1 / n k ≥ c > 1 ∑ k 1 F n k \sum_k\frac{1}{F_{n_k}} ∑ k F n k 1
Is it true that if a 1 < a 2 < ⋯ a_1<a_2<\cdots a 1 < a 2 < ⋯ lim inf a n 1 / 2 n > 1 \liminf a_n^{1/2^n}>1 lim inf a n 1/ 2 n > 1 ∑ n = 1 ∞ 1 a n a n + 1 \sum_{n=1}^\infty \frac{1}{a_na_{n+1}} ∑ n = 1 ∞ a n a n + 1 1
Is … irrational, where μ \mu μ
Let t > 1 t>1 t > 1 ∑ n = 1 ∞ 1 t n − 1 = ∑ n = 1 ∞ τ ( n ) t n \sum_{n=1}^\infty\frac{1}{t^n-1}=\sum_{n=1}^\infty \frac{\tau(n)}{t^n} ∑ n = 1 ∞ t n − 1 1 = ∑ n = 1 ∞ t n τ ( n ) τ ( n ) \tau(n) τ ( n ) n n n
Let f : N → N f:\mathbb N→\mathbb N f : N → N f ( n ) → ∞ f(n)→∞ f ( n ) → ∞ ∑ n ≥ 1 ( ( n + 1 ) ( n + 2 ) ⋯ ( n + f ( n ) ) ) − 1 ∑_{n≥1}((n+1)(n+2)⋯(n+f(n)))^{-1} ∑ n ≥ 1 (( n + 1 ) ( n + 2 ) ⋯ ( n + f ( n )) ) − 1
Let P P P ∣ P ∣ ≥ 2 \lvert P\rvert \geq 2 ∣ P ∣ ≥ 2 { a 1 < a 2 < ⋯ } = { n ∈ N : if p ∣ n then p ∈ P } \{a_1<a_2<\cdots\}=\{ n\in \mathbb{N} : \textrm{if }p\mid n\textrm{ then }p\in P\} { a 1 < a 2 < ⋯ } = { n ∈ N : if p ∣ n then p ∈ P }
Is the following universal assertion false? For every sequence a : N → N a:\mathbb{N}\to\mathbb{N} a : N → N a ( n ) ≥ 1 a(n)\geq1 a ( n ) ≥ 1 n n n
Let 1 ≤ a 1 < a 2 < ⋯ 1\leq a_1<a_2<\cdots 1 ≤ a 1 < a 2 < ⋯ a n → ∞ a_n\to \infty a n → ∞ ∑ 1 a n and ∑ 1 a n − 1 \sum\frac{1}{a_n}\quad\textrm{and}\quad\sum\frac{1}{a_n-1} ∑ a n 1 and ∑ a n − 1 1
Let a n a_n a n b n b_n b n a n + b n ≠ 0 a_n+b_n\neq 0 a n + b n = 0 b n ≠ 0 b_n\neq 0 b n = 0 n n n
Let a n a_n a n b n b_n b n b n / a n → 1 b_n/a_n\to 1 b n / a n → 1 ∑ 1 b n \sum\frac{1}{b_n} ∑ b n 1
Call a 1 < a 2 < ⋯ a_1<a_2<\cdots a 1 < a 2 < ⋯ ∑ n ≥ 1 1 / ( t n a n ) \sum_{n\geq1}1/(t_na_n) ∑ n ≥ 1 1/ ( t n a n ) t n ≥ 1 t_n\geq1 t n ≥ 1
Let a 1 < a 2 < ⋯ a_1<a_2<\cdots a 1 < a 2 < ⋯ a n / n → ∞ a_n/n\to \infty a n / n → ∞ ∑ n a n 2 a n \sum_n \frac{a_n}{2^{a_n}} ∑ n 2 a n a n
Let A ⊆ N A\subseteq \mathbb{N} A ⊆ N ∑ n ∈ A 1 2 n − 1 \sum_{n\in A}\frac{1}{2^n-1} ∑ n ∈ A 2 n − 1 1
Let k ≥ 1 k\geq 1 k ≥ 1 σ k ( n ) = ∑ d ∣ n d k \sigma_k(n)=\sum_{d\mid n}d^k σ k ( n ) = ∑ d ∣ n d k ∑ σ k ( n ) n ! \sum \frac{\sigma_k(n)}{n!} ∑ n ! σ k ( n )
Is ∑ p n 2 n \sum \frac{p_n}{2^n} ∑ 2 n p n p n p_n p n n n n
Is it true that every real number x x x σ ( n ) \sigma(n) σ ( n )
Is ∑ n ϕ ( n ) 2 n \sum_n \frac{\phi(n)}{2^n} ∑ n 2 n ϕ ( n ) ϕ \phi ϕ
Let 1 ≤ a 1 < a 2 < ⋯ 1\leq a_1<a_2<\cdots 1 ≤ a 1 < a 2 < ⋯ lim sup a n n = ∞ . \limsup \frac{a_n}{n}=\infty. lim sup n a n = ∞. ∑ n = 1 ∞ 1 2 a n \sum_{n=1}^\infty \frac{1}{2^{a_n}} ∑ n = 1 ∞ 2 a n 1
Is the real number … irrational, where ω ( n ) \omega(n) ω ( n ) n n n
Is ∑ n ≥ 2 1 n ! − 1 \sum_{n\geq 2}\frac{1}{n!-1} ∑ n ≥ 2 n ! − 1 1
Monotone subsequences conjecture. There exists a subsequence of the first sequence that is monotone non-decreasing and a subsequence of the second sequence that is monotone non-inc…
Product-of-curves conjecture.
Stapleton's asymptotic conjecture. The degree of irrationality of a very general polarized K 3 K3 K 3 d \sqrt d d
