# Tagged Questions

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### Help with integral/logarithm inequality

I have to prove the following inequality: $1/(n+1) < \int_n^{n+1} 1/t$ $dt$ $<1/n$ I thought it would be easier to attack this via integration, so I get: $1/(n+1) <$ log $(n+1)-$ ...
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### Solving inequality having log

I am struggling to solve this inequality involving logarithm. How to find out values of $n$ for which below inequality holds good: $${\log_2n \over n} >{ 1 \over 8}$$
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### Inequality $C\lceil\log{n}\rceil! \geq n^k$

I've been struggling to prove there exist $C$ for $n, n_{0}, \forall k >0 \in \mathbb{R}$ such that $\forall n > n_{0}$: $$C\lceil\log{n}\rceil! \geq n^k$$ As you ...
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### Inequality with logarithms

How do I show that $$\frac{1}{n-1}\geq \ln \left ( \frac{n}{n-1} \right )$$ for $n>1$? As far as I can tell, exponentiating both sides with base $e$ won't help, because then I get a nasty ...
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### determine x in $x\log_\frac{1}{10}(x^2+x+1)>0$

I wanted to know, how can i determine the values of x for which $x\log_\frac{1}{10}(x^2+x+1)>0$ going to the question, we must have $x>0$ and $\log_\frac{1}{10}(x^2+x+1)>0$ or both must ...
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### If $ax^2-bx+c=0$ has two distinct real roots lying in the interval $(0,1)$ $a,b,c$ belongs to natural prove that $\log_5 {abc}\geq2$

If $ax^2-bx+c=0$ has two distinct real roots lying in the interval $(0,1)$ with $a, b, c\in \mathbb N$, prove that $\log_5 {abc}\geq2$. The equations I could form are: 1) $f(0)>0$ and ...
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### Do these inequalities regarding the gamma function and factorials work?

I am seeing if it is possible to generalize lemma 1 in Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$. In a previous question, I asked whether the following inequality is ...
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### Trying to generalize an inequality from Jitsuro Nagura: Does this work?

I am investigating the generality of Lemma 1 in Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$. In Lemma 1, Nagura establishes when $n > 1$, $x \ge 1$: ...
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### I am trying to solve the inequality $\log_{\log{\sqrt{9-x^2}}} x^2 <0$

I am trying to solve the inequality $$\log_{\log{\sqrt{9-x^2}}} x^2 <0.$$ I got $\mathrm{S.S}=(-\sqrt8 ,-1)\cup( 1,\sqrt8)$, but a friend got $\mathrm{S.S}=(-1,1)- \{0\}$. Please, what is ...
Is there a graphic visualization of $\sum_{k=1}^{n} 1/k \, \, \leq \, \, \,1 \, + \, \int_1^n \! (1/x) \, \mathrm{d} x$ as intuitive as the integral test ? I can't see why the inequality is true. I ...
I need a function $f(x)$ that satisfies the properties bellow for all integers $k$ $$\frac{\log(k+1)}{k+1}-\log\left(1+\frac 1 k\right)+f(k+1)-f(k)<0 \$$ $$\lim_{k \rightarrow \infty} f(k)=0$$ ...