Tagged Questions
0
votes
2answers
103 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
63 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
36 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
433 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 + ...
4
votes
1answer
499 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 ...
4
votes
1answer
237 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
273 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 ...
15
votes
6answers
831 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 ...
