# Tagged Questions

Questions on proving, manipulating and applying inequalities.

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### How to prove that $\log(x)<x$ when $x>1$?

It's very basic but I'm having trouble to find a way to prove this inequality $\log(x)<x$ when $x>1$ ($\log(x)$ is the natural logarithm) I can think about the two graphs but I can't find ...
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### Comparing $\pi^{e}$ and $e^{\pi}$

How can I calculate without calculator or something like this the values of $\pi^{e}$ and $e^{\pi}$ in order to compare them ?
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### how to strictly prove $\sin x<x$ for $0<x<\frac{\pi}{2}$

$$\sin x<x\,(0<x<\frac{\pi}{2})$$ In most textbooks, to prove this inequality is based on geometry illustration (draw a circle, compare arc length and chord ), but I think that strict proof ...
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### How to prove $a^2 + b^2 + c^2 \ge ab + bc + ca$?

How can the following inequation be proven? $$a^2 + b^2 + c^2 \ge ab + bc + ca$$
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### How to prove the inequality $2\sqrt{n + 1} − 2 \le 1 +\frac 1 {\sqrt 2}+\frac 1 {\sqrt 3}+ \dots +\frac 1 {\sqrt n} \le 2\sqrt n − 1$?

Prove that for any positive integer $n$, $$2\sqrt{n + 1} − 2 \le 1 +\frac 1 {\sqrt 2}+\frac 1 {\sqrt 3}+ \dots +\frac 1 {\sqrt n} \le 2\sqrt n − 1$$ Progress I think Riemann sum should be used for ...
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### How do you prove ${n \choose k}$ is maximum when k is $\lceil \frac n2 \rceil$ or $\lfloor \frac n2\rfloor$?

How do you prove ${n \choose k}$ is maximum when k is $\lceil \frac{n}{2} \rceil$ or $\lfloor \frac{n}{2} \rfloor$ ? This link provides a proof of sorts but it is not satisfying. From what I ...
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### Prove that $n^k < 2^n$ for all large enough $n$

If $k\ge 2$ is an integer, prove by elementary means (no calculus or limits) that there is a $N(k)$ such that $n^k < 2^n$ for all integers $n \ge N(k)$. Give an explicit form for $N(k)$.
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### A series to prove $\frac{22}{7}-\pi>0$

After T. Piezas answered Is there a series to show $22\pi^4>2143\,$? a natural question is Is there a series that proves $\frac{22}{7}-\pi>0$? One such series may be found combining ...
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### What does $\ll$ mean?

I saw two less than signs on this Wikipedia article and I was wonder what they meant mathematically. http://en.wikipedia.org/wiki/German_tank_problem EDIT: It looks like this can use TeX commands. ...
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### Factorial Inequality problem $\left(\frac n2\right)^n > n! > \left(\frac n3\right)^n$

I met an inequality, I ask, do not mathematical induction to prove that: Prove $\left(\frac n2\right)^n > n! > \left(\frac n3\right)^n$ without using induction
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### Proving $1+\frac{1}{4}+\frac{1}{9}+\cdots+\frac{1}{n^2}\leq 2-\frac{1}{n}$ for all $n\geq 2$ by induction

Question: Let $P(n)$ be the statement that $1+\dfrac{1}{4}+\dfrac{1}{9}+\cdots +\dfrac{1}{n^2} <2- \dfrac{1}{n}$. Prove by mathematical induction. Use $P(2)$ for base case. Attempt at ...
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### Series and integrals for inequalities and approximations to $\pi$

Fundamentals Two beautiful expressions that relate $\pi$ to its convergents are Dalzell integral $$\frac{22}{7}-\pi=\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx$$ (see Why do we need an integral to prove ...
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### Prove $\left(\dfrac{2}{5}\right)^{\frac{2}{5}}<\ln{2}$

Inadvertently, I find this interesting inequality,But this problem have nice solution? prove that $$\ln{2}>(\dfrac{2}{5})^{\frac{2}{5}}$$ This problem have nice solution? Thank you. ago,I find ...
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### Fastest way to check if $x^y > y^x$?

What is the fastest way to check if $x^y > y^x$ if I were writing a computer program to do that? The issue is that $x$ and $y$ can be very large.
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### How prove this inequality $(a^2+bc^4)(b^2+ca^4)(c^2+ab^4) \leq 64$

Question: If $a,b,c$ are nonnegative real numbers such that $a+b+c=3,$ then $$(a^2+bc^4)(b^2+ca^4)(c^2+ab^4) \leq 64$$ My try: I found this inequality is only if $(a,b,c)=(2,0,1)$ But I can'...
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### Proving $(1 + 1/n)^{n+1} \gt e$

I'm trying to prove that $$\left(1 + \frac{1}{n}\right)^{n+1} > e$$ It seems that the definition of $e$ is going to be important here but I can't work out what to do with the limit in the ...
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### $a+b+c =0$; $a^2+b^2+c^2=1$. Prove that: $a^2 b^2 c^2 \le \frac{1}{54}$

If a,b,c are real numbers satisfying $a+b+c =0; a^2+b^2+c^2=1$. Prove that $a^2 b^2 c^2 \le \frac{1}{54}$.
### Maximum value of $\sin A+\sin B+\sin C$?
What is the maximum value of $\sin A+\sin B+\sin C$ in a triangle $ABC$. My book says its $3\sqrt3/2$ but I have no idea how to prove it. I can see that if $A=B=C=\frac\pi3$ then I get \$\sin A+\sin ...