4
votes
1answer
139 views

What $\alpha$ such that if $xy=\alpha$, then $e^{-x}+e^{-y}\geq 2e^{-\sqrt \alpha} $?

For every $ x,y \gt 0$, if $ xy=\alpha$, then we have $$e^{-x}+e^{-y}\geq 2e^{-\sqrt \alpha} $$ What are the possible values of $\alpha$? $2 < e^{1/(n+1)} + e^{-1/n}$ led to this problem. ...
7
votes
2answers
92 views

Is $\int_x^{\infty}e^{-\frac{t^2}{2}} < \frac{1}{x}e^{-\frac{x^2}{2}}$?

While solving a problem in real analysis, I got stuck. I need to prove $$\int_x^{\infty}e^{-\frac{t^2}{2}}dt < \frac{1}{x}e^{-\frac{x^2}{2}} $$ Clearly I have to use some kind of inequality, but ...
1
vote
2answers
52 views

An exponential/polynomial inequality

Prove that there is at least $1$ real number $a>0$ with the property $$a^x\ge x^a $$ for any $x>0$.
8
votes
3answers
122 views

Showing $n!<e(\frac{n}{2})^n$

I'd like to prove that $n!<e(\frac{n}{2})^n$. What I have so far: $\sqrt[n]{n!} = \sqrt[n]{1\cdot 2 \cdot \ldots \cdot n} \leq \frac{1+\ldots +n}{n}=\frac{(n+1)n}{2n}=\frac{(n+1)}{2}$. Thus ...
2
votes
3answers
647 views

How to solve exponential function inequality?

How do I solve the exponential equations like $2^\frac{x}{8}<x$? I can solve this by plotting into graph. But is there any way to do it mathematically? or like $2^x < 100x^2$ . I am trying to ...
18
votes
4answers
304 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 ...
2
votes
6answers
288 views

If $a_1,a_2,\ldots,a_n>0$ and $a_1+a_2+\ldots+a_n<\frac{1}{2}$, prove that $(1+a_1)(1+a_2)\ldots(1+a_n)<2$.

If $a_1,a_2,\ldots,a_n>0$ and $a_1+a_2+\cdots+a_n<\frac{1}{2}$, prove that $$(1+a_1)(1+a_2)\cdots(1+a_n)<2$$ I've tried using Holder's inequality (the same result can easily be derived using ...
1
vote
1answer
44 views

Outputting inequality with $e^x$

I many books I can find inequality which estimates $e$: $$\left(1+\frac{1}{n}\right)^n \lt e \lt \left(1+\frac{1}{n}\right)^{n+1}$$ I am wondering if correct is also to write: ...
0
votes
1answer
44 views

Derivation of this inequality [duplicate]

Graphically, it is easy to see but to analytically derive the following inequality, $$|e^x-e^y| \le |x-y| \ \ \ \ \forall x,y \in (-\infty,0] \ \ \ \ (1)$$ do I just apply the MVT and show that ...
0
votes
1answer
63 views

The least value of $b$ such that $2^{3^{\cdots^a}}\leq b^{(b-1)^{\cdots^2}}$

I'm interested in an upper bound for the minimal positive integer $b$ for which $$2^{3^{4^{\cdots^a}}}\leq b^{(b-1)^{\cdots^{3^{2}}}}$$ holds, given a positive integer $a\geq2$. If possible, but I ...
0
votes
1answer
32 views

Question about $\Sigma_{i=1}^n (a_i^z-a_i^{-z})$

Let $z$ be a complex number and $n$ a positive integer. Let $a_n$ be a sequence of $n$ real numbers such that $a_n > 1$ for every $n$. Define $f_n(z;a_1,a_2,...,a_n)=\Sigma_{i=1}^n ...
1
vote
1answer
82 views

question about an inequality in calculus [duplicate]

Please, carefully show that $$ e^{\pi} > \pi^e $$ You are not allowed to use a calculator! thanks
5
votes
0answers
110 views

How prove this inequality $\frac{1}{n!}\sum\limits_{k=0}^{\infty}\frac{k^n}{k!}\ge e(C\ln{n})^{-n}$

Show that: $$\dfrac{e^n}{(\ln{n})^n}\ge \dfrac{1}{n!}\sum_{k=0}^{\infty}\dfrac{k^n}{k!}\ge e(C\ln{n})^{-n},\ n\ge 2$$ where $C>e$ is constant. My try: ...
2
votes
4answers
79 views

Prove $e^x \geq x+1$

I've tried finding a solution to this problem. I want to use the following definition of e $e = \lim_{n \to\infty} (1+1/n)^n$ I have seen people argue with Bernoulli's Inequality, saying $1+x \leq ...
0
votes
1answer
68 views

Solve the inequality $2^{\left( x^{3}-x\right) } < 1$

$2^{\left( x^{3}-x\right) } < 1$ Let $2^{\left( x^{3}-x\right) }-1=f\left( x\right)$ To find the values for which $f(x)<0$ I let $f(x)=0$: $2^{\left( x^{3}-x\right) }-1=0$ $2^{\left( ...
1
vote
4answers
246 views

What method can obtain value of $d,$ when $d$ is an exponent?

I am a beginner in math. How I find the value of d, when $d$ is an exponent? What method can solve it? $$m_o > 2^d$$
5
votes
2answers
37 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 ...
5
votes
2answers
161 views

Upper bound for $(1-1/x)^x$

I remember the bound $$\left(1-\frac1x\right)^x\leq e^{-1}$$ but I can't recall under which condition it holds, or how to prove it. Does it hold for all $x>0$?
3
votes
1answer
85 views

Prove that $pe^{\alpha q} + qe^{- \alpha p} \le e ^ {\alpha^2/2}$

Prove that, $$pe^{\alpha q} + qe^{- \alpha p} \le e ^ {\alpha^2/2}$$ where $p$ and $q$ are the probabilities of success and failure in a Bernoulli trial ($0 \le p \le 1, 0 \le q \le 1, p + q = 1$) ...
28
votes
18answers
2k views

Simplest or nicest proof that $1+x \le e^x$

The elementary but very useful inequality that $1+x \le e^x$ for all real $x$ has a number of different proofs, some of which can be found online. But is there a particularly slick, intuitive or ...
0
votes
1answer
58 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 ...
1
vote
1answer
108 views

Proving that linear combination of exponentials is positive

I found the following question in a book without any proof. Question : Prove that $$f(t)=3-5e^{-2t}+6e^{-3t}+2e^{-5t}-3e^{-(3-\sqrt5)t}-3e^{-(3+\sqrt5)t}\gt0$$ for any $t\gt0$. The book says that ...
2
votes
1answer
140 views

How can I prove the following exponential inequality?

I'm trying to prove the following inequality with no success: $$ e^{-\frac{1}{x}}e^{-\frac{1}{y}}e^{-\frac{1}{x+y}} \leq e^{-\frac{1}{x}}+e^{-\frac{1}{y}}-e^{-\frac{1}{x+y}} $$ for $x>1$, and ...
6
votes
3answers
124 views

How to prove $(\frac{n+1}{e})^n<n!<e(\frac{n+1}{e})^{n+1}$ without integrating method?

How to prove $$\left(\frac{n+1}{e}\right)^n<n!<e\left(\frac{n+1}{e}\right)^{n+1}$$ without integrating method? In fact we could prove this by noticing that $$i<x<i+1\Rightarrow \ln ...
1
vote
0answers
41 views

Inequality of Partial Taylor Series

For a given $\theta < 1$, and $N$ a positive integer, I am trying to find an $x > 0$ (preferably the smallest such $x$) such that the following inequality holds: $$\sum_{k=0}^{N} \frac{x^k}{k!} ...
1
vote
3answers
92 views

Infinite Series

How can you show that $$\left(1-\frac{2}{n^2}\right)^{n^2/2} \le \frac{1}{e}\:\: \qquad\forall n \ge 2$$ Any ideas? Infinite series have never really been my thing. Thanks
3
votes
2answers
85 views

Inequality with a sum

I am reading Remarks on a Ramsey theory for trees: Janos Pach, Jozsef Solymosi, Gabor Tardos http://arxiv.org/abs/1107.5301 I am stuck at inequality in proof of Lemma 6. $n\geq 8$, $k=2\lfloor ...
9
votes
4answers
235 views

prove that $\frac{1-e^{-x^2}}{x}\le 2\sqrt{2} , \ x>0$,

Can you show very easy methods? I hope I'll see many methods. Thank you everyone. Prove that: $$\frac{1-e^{-x^2}}{x}\le 2\sqrt{2} \ \ \ \qquad \forall x>0.$$
10
votes
2answers
202 views

Prove that $(e+x)^{e-x}>(e-x)^{e+x}$

I get stuck with proving that $$(e+x)^{e-x}>(e-x)^{e+x}$$ for $x \in (0, e)$. All I know, is that it is doable with Jensen inequality, and I started with defining $$f(x)=(e+x)^{e-x}$$ and further ...
7
votes
2answers
129 views

How do I prove that $\exp(\frac{h}{1+h})\leq 1+h$?

I have come across the inequality $$\exp\left(\frac{h}{1+h}\right)\leq 1+h,\quad\forall h>-1,$$ on http://functions.wolfram.com/ElementaryFunctions/Exp/29/. I would like some help proving this. A ...
0
votes
1answer
45 views

How to solve the following special inequality?

Find $k$, as a function of $d_2$ and $d_3$, such that: $$\left \vert { d_2 \left [ \sin(e^{d_3\,y}) - \sin(e^{d_3\,x})\right] + (x-y) d_2 d_3 e^{d_3\,z} \cos(e^{d_3\,z})} \right \vert \le k ...
6
votes
4answers
711 views

proof of inequality $e^x\le x+e^{x^2}$

Does anybody have a simple proof this inequality $$e^x\le x+e^{x^2}.$$ Thanks.