# Tagged Questions

Questions on proving, manipulating and applying inequalities.

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### How to prove $a,b,c \in \mathbb{R} \mid a+b+c \geq abc \implies 3abc(a+b+c) \geq 3(abc)^2$?

I'm working through some proofs from Cvetkovski's "Inequalities", when I came across a more difficult one (for newbies like me). Given $a, b, c \in \mathbb{R} \mid a+b+c \geq abc$, how can we prove ...
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### How to compare $\left(\sin \left(x\right)\right)^{\cos \left(x\right)}$ and $\left(\cos \left(x\right)\right)^{\sin \left(x\right)}$

I am new here ,can anybody help to solve this problem: How to compare $\left(\sin \left(x\right)\right)^{\cos \left(x\right)}$ and $\left(\cos \left(x\right)\right)^{\sin \left(x\right)}$ in the ...
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### The 1000th partial sum of the series $\sum 1/n^2$ is less than 2 [closed]

Can anyone help me with this problem: Prove: $$1/1^2 + 1/2^2 +1/3^2 +\dots +1/1000^2 <2$$
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### Can this upper bound for $\sum_{i=1}^n \lfloor \sqrt{p_i} \rfloor, p_i \in \Bbb P$ be improved?

I would like to find the smallest possible upper bound for the following sum of prime radicals (OEIS A062048): $\sum_{i=1}^n \lfloor \sqrt{p_i} \rfloor, p_i \in \Bbb P$ This is my attempt. It ...
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If $f(x)=x\ln x-\frac{k}{x}$. And $x_1$, $x_2$ are two roots of $f(x)=0$. Then $f'\left(\frac{x_1+x_2}{2}\right)\not=0$ First, I determine the range of $k$. Because $f=0$ has two roots $\iff$ $x^2\... 3answers 53 views ### Suppose that$(s_n)$converges to$s$,$(t_n)$converges to$t$, and$s_n \leq t_n \: \forall \: n$. Prove that$s \leq t$. I'm stuck with the proof of the following: Suppose that$(s_n)$converges to$s$,$(t_n)$converges to$t$, and$s_n \leq t_n \: \forall \: n$. Prove that$s \leq t$. I've tried starting with$s_n \...
Let $z$ be a complex number with $\mathrm{Im}(z)>0$, and we consider $$w:=\frac{-z+\sqrt{z^2-4}}{2}.$$ It is written that "we take the square root so that $\mathrm{Im}(w)>0".$ I want to prove ...