# Tagged Questions

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### Inequality: $\left|x^3-y^3\right|<|x|^3+|y|^3$

Could anyone show me why $$\left|x^3-y^3\right|<|x|^3+|y|^3$$ for all real numbers (x,y) except 0? I'm thinking of whether of how to remove the modulus sign on the left hand side of the ...
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### pow$(X,Y)$ $>$ pow$(Y,X)$, if $X<Y$.

How can we proof following? if $X < Y$, then: $X^{Y} > Y^{X}$ , Where X, and Y are integers. Also $X,Y > 1$. Except a special case $2^{3} < 3^{2}$. I think for other ...
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### Summation of powers inequality

Can anyone provide a slick proof of the following? Let $0 < x \le 1$. Then $\displaystyle \sum_{k=0}^{n-1} x^k \ge \frac {1} {1 - (1 - 1/n)x}$.
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### Problem understanding a proof about powers in ordered fields

I am reading through a textbook on Analysis and have come across a question that I can't seem to make any headway with. A proof is outlined, but I can't make any sense out of it. The problem is as ...
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### Prove by induction that for all $n \geq 3$: $n^{n+1} > (n+1)^n$

I am currently helping a friend of mine with his preperations for his next exam. A big topic of the exam will be induction, thus I told him he should practice this a lot. As at the beginning he had no ...
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### Comparing Powers of Different Bases

How can I know if one power is bigger than the other when the bases are different? For example, considering $2^{10}$ and $10^{3}$ the former is the greater one, but how to prove this? Logarithms? ...
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### Power function inequality

Let $x$ and $p$ be real numbers with $x \ge 1$ and $p \ge 2$ . Show that $(x - 1)(x + 1)^{p - 1} \ge x^p - 1$ . I recently discovered this result. I am sure it is known, but it is new to me. It is ...
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### Solving an exponential inequality

$$(0{,}25)^{3-0{,}5x^2}\leq8$$ Answers given are: $[-3;3]$ Below is where I got with this, I'm pretty sure I took a wrong approach here. Any help at all is appreciated. \begin{aligned} ...
### Simple estimation $e^{a\sqrt{r}} > r$
I want to prove a simple theorem about contour integration via residues and I need the following estimation: $e^{a\sqrt{r}} > r$ for any real a > 0 and r >> 0. Is this true? If so, what is an ...