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Mathematics
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WuKong
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36
Let $a_1=\sqrt{6},a_{n+1}=\sqrt{6+a_n}$. Find $\lim\limits_{n \to \infty} (a_n-3)6^n$.
Apr 11, 2021
31
Prove or disprove that, for any $n \in \mathbb{N_+}$, there exist $a,b \in \mathbb{N_+} $ such that $\frac{a^2+b}{a+b^2}=n.$
May 31, 2018
28
Why does Wolfram|Alpha make a mistake here?
Oct 8, 2018
26
Evaluate $\lim\limits_{n\rightarrow \infty}\frac{n+n^2+n^3+\cdots +n^n}{1^n+2^n+3^n+\cdots +n^n}.$
May 23, 2018
22
Prove or disprove that $2^{\frac{\pi}{2}}$ is an irrational number.
May 25, 2018
21
Find $\lim_{n \to \infty}\frac{x_n}{\sqrt{n}}$ where $x_{n+1}=x_n+\frac{n}{x_1+x_2+\cdots+x_n}$
Jul 8, 2022
19
Prove that $a^4+b^4+1\ge a+b$.
Jul 20, 2018
18
Show that $\int_a^b \sin\left(x+\frac{1}{x}\right) dx <3.$
Jul 1, 2021
15
Evaluate $\lim\limits_{n \to \infty}\frac{(-1)^n}{n!}\int_0^n (x-1)(x-2)\cdots(x-n){\rm d}x$
Jul 4, 2020
14
Find $\lim\limits_{n \to \infty}\sum_{k=1}^n\left(\frac{k}{n}\right)^k$.
Mar 24, 2020