1
vote
2answers
34 views

Inequality with trigonometric functions

Find all values for $a$ such that the following inequality holds: $$\sin^6x + \cos^6x + a\sin x \cos x \ge 0$$ To be fair, I didn't manage to get anything helpful wiht my calculations. I tried to ...
1
vote
1answer
18 views

Supremum of (e^(i z t) - 1)/z

i'm new here, so i'm not sure if this is the right place to ask this question: I know that the following holds true: $$ \forall\, t \in \mathbb{R} \; \forall\,x\in\mathbb{R}\setminus\{0\} ...
18
votes
4answers
287 views

Find the smallest k such that $n^k > \sum_{i=0}^{n-1} i^k$

Let $n \in \mathbb{N}$. Is it possible to find the smallest $k \in \mathbb{N}$ such that $$n^k > \sum_{i=1}^{n-1} i^k \ ?$$ It's easy to prove that such $k$ exist because: $$n^k > 1^k + 2^k ...
0
votes
1answer
35 views

Prove that $\forall \, a,b \in \mathbb{N}- \{0,1\}\,\, \wedge \,\,a<b \,\, ; \,\, a^{1/a} > b^{1/b}$

Prove that $\forall \, a,b \in \mathbb{N}- \{0,1\}\,\, \wedge \,\,a<b \,\, ; $ $$\,\, a^{1/a} > b^{1/b}$$ I need some tip to start it. Thank you.
0
votes
0answers
65 views

difficult inequality to prove

I need help proving this inequality is correct for a homework assignment: $$\displaystyle \left(\frac{13}{4}\right)^{n} \leq ...
1
vote
4answers
145 views

Which is larger :: $y!$ or $x^y$, for numbers $x,y$.

This is a generalization of this question :: Which is larger? $20!$ or $2^{40}$?. No explicit general solution was presented there and I'm just curious :D Thank-you. Edit :: I want a most-general ...
5
votes
2answers
33 views

Upper bound for product of exponents

From here we have the bound $$\left(1-\frac1N\right)^N\leq e^{-1}$$ where $N$ is a positive integer. Written another way, it is ...
6
votes
1answer
150 views

About $e^{\pi}\gt {\pi}^e, \ e^{e^{\pi}}\lt {\pi}^{{\pi}^{e}},e^{{\pi}^{{\pi}^e}}\gt {\pi}^{e^{e^{\pi}}}$ and their generalization

Let us define a sequence $\{(a_n,b_n)\}$ as $$(a_1,b_1)=(e,\pi),\ \ (a_{2n},b_{2n})=(e^{b_{2n-1}},{\pi}^{a_{2n-1}}),\ \ (a_{2n+1},b_{2n+1})=(e^{a_{2n}},{\pi}^{b_{2n}})$$ for $n=1,2,3,\cdots$. Then, ...
40
votes
2answers
2k views

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.
1
vote
2answers
78 views

Checking inequality without actually calculating LHS and RHS

How to check whether the following inequality is true or not without actually calculating the values of $x^y $ and $y^x $: $$ x^y > y^x$$ (x and y are integers)
1
vote
2answers
31 views

threshold of n to satisfy $a^n <n^a$

How to find the minimum of $n$ when we know $a$, to satisfy: $a^n<n^a$ $a^m>m^a$ for each $m>n$ $n$ and $m$ are natural numbers.
0
votes
1answer
55 views

Relation of $e$ to other numbers…

I found the following result, When i was working on my calculator . $$x^y < y^x \quad ,x < y \quad \text{ for } x,y<e$$ $$x^y > y^x \quad ,x < y \quad \text{ for } x,y>e$$ I can't ...
6
votes
0answers
178 views

Close power towers

In another question I gave some ways to determine which of two power towers is larger, but my answer there is incomplete because it doesn't handle the case where two towers are very close at each ...
7
votes
1answer
110 views

Power tower inequality

I want to prove the following power tower inequality: $$ 3 \uparrow \uparrow 100 > 4 \uparrow \uparrow 99 $$ but I don't know how to do this. I think that induction will not work, because I think ...
2
votes
2answers
272 views

Do inequalities hold under square-root (or exponentiation in general)?

This has been bothering me lately. My proof-skills are rusty (and were never great to begin with). I dimly recall having seen this (or something related to it) in a math course I took a while ago, but ...
1
vote
1answer
78 views

A simple inequality with logarithms and exponential

I want to prove that for $k>0$: $ 2^k \geq \frac{-1}{\log_2(1-\frac{1}{2^k})}$ I've plotted both functions and it seems to be the case for k>0. In fact, it would also be nice to see that: $ ...
0
votes
2answers
145 views

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 ...
3
votes
1answer
155 views

Summation of powers inequality

Can anyone provide a slick proof of the following? Let $0 < x \le 1$. Then $\displaystyle \sum_{k=0}^{n-1} x^k \ge \frac {1} {1 - (1 - 1/n)x}$.
1
vote
1answer
47 views

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 ...
3
votes
2answers
866 views

The inequality $b^n - a^n < (b - a)nb^{n-1}$

I'm trying to figure out why $b^n - a^n < (b - a)nb^{n-1}$. Using just algebra, we can calculate $ (b - a)(b^{n-1} + b^{n-2}a + \ldots + ba^{n-2} + a^{n-1}) $ $ = (b^n + b^{n-1}a + \ldots + ...
12
votes
6answers
1k views

Proving the inequality $e^{-2x}\leq 1-x$

How do I prove the inequality $e^{-2x}\leq1-x$ for $0\leq x\leq1/2$?
2
votes
1answer
195 views

Justifying a pair of inequalities involving the exponential function

I'm reading Fan Chung's Spectral Graph Theory. There's a pair of inequalities I don't know how to justify. Chung doesn't attempt to explain them, so maybe they're very obvious. Example 1.19 on page ...
5
votes
1answer
900 views

Prove variant of triangle inequality containing p-th power for 0 < p < 1

Sorry if this is a trivial question, but I am kind of stuck with proving the following inequality and have been searching for a while: $\rho \left( \sum\limits_i^n d_i \right) \leq \sum\limits_i^n ...
9
votes
10answers
311 views

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 ...
5
votes
1answer
412 views

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? ...
4
votes
3answers
351 views

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 ...
1
vote
3answers
444 views

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} ...
1
vote
1answer
83 views

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 ...
17
votes
7answers
954 views

Simple proof that $8\left(\frac{9}{10}\right)^8 > 1$

This question is motivated by a step in the proof given here. $\begin{align*} 8^{n+1}-1&\gt 8(8^n-1)\gt 8n^8\\ &=(n+1)^8\left(8\left(\frac{n}{n+1}\right)^8\right)\\ &\geq ...
0
votes
1answer
158 views

Proof that $a^x$ < $a^{x + \Delta x}$

How can I prove that $a^x$ < $a^{x + \Delta x}$, where $\Delta x > 0$, $a$ is a constant, and $a > 1$ (in my case, $a=2$)? I don't want to use a graph, of course.