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Omni-MATH / For a real number r, set ‖r‖=min{|r−n|:n∈Z}, where |⋅| means the absolute value of a real number. 1. Is there a nonzero real number s, such that lim_(n→∞)‖(√(2)+1)^(n)s‖=0 ? 2. Is …

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problem

For a real number rr, set ‖r‖=min{|r−n|:n∈Z}\|r\|=\min \{|r-n|: n \in \mathbb{Z}\}, where |⋅||\cdot| means the absolute value of a real number. 1. Is there a nonzero real number ss, such that limn→∞‖(2+1)ns‖=0\lim _{n \rightarrow \infty}\left\|(\sqrt{2}+1)^{n} s\right\|=0 ? 2. Is there a nonzero real number ss, such that limn→∞‖(2+3)ns‖=0\lim _{n \rightarrow \infty}\left\|(\sqrt{2}+3)^{n} s\right\|=0 ?
Plain-text mathematical notation (without MathML)
For a real number r, set ‖r‖=min{|r−n|:n∈Z}, where |⋅| means the absolute value of a real number. 1. Is there a nonzero real number s, such that lim_(n→∞)‖(√(2)+1)^(n)s‖=0 ? 2. Is there a nonzero real number s, such that lim_(n→∞)‖(√(2)+3)^(n)s‖=0 ?
Original LaTeX notation
For a real number $r$, set $\|r\|=\min \{|r-n|: n \in \mathbb{Z}\}$, where $|\cdot|$ means the absolute value of a real number. 1. Is there a nonzero real number $s$, such that $\lim _{n \rightarrow \infty}\left\|(\sqrt{2}+1)^{n} s\right\|=0$ ? 2. Is there a nonzero real number $s$, such that $\lim _{n \rightarrow \infty}\left\|(\sqrt{2}+3)^{n} s\right\|=0$ ?

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task_id
1df8d3ac-8f04-5446-baef-066655f4245d
task_key
test--1df8d3ac-8f04-5446-baef-066655f4245d
task_revision_id
1
config / split
/ test

Official source

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