Show that $$\frac{1}{2}&lt;\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n^p}&lt;1$$ for all positive number $p$. And the given hint is that $f(x)=\frac{1}{x^p}$ is convex for $x&gt;0$.  I can find the upper bound 1, but have no ideas for its lower bound. How can I exploit the convexity of $f(x)=\frac{1}{x^p}$?
121563
Let $p&gt;1$. Prove that the series:  $\sum_{n=1}^\infty \frac {(-1)^{n+1}}{n^p}$  is between $\frac {1}{2}$ and 1.  Any help is appreciated.  Just a challenge problem I was presented and curious on the solution  Thanks
74920
Suppose that $0\neq x\in\mathbb{R}$ and $x + \frac1x\in\mathbb{Z}$. Prove that, for all  $n\ge1$, $x^n + \frac1{x^n}\in\mathbb{Z}$.  I can't figure out and understand the question. Can you give me some hints ?
90967
1) Let $F_1 = 1$, $F_2 = 1$, and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 3$.   How do you prove that  $$\sum_{n=2}^\infty \frac1{F_{n-1} F_{n+1}} = 1$$   $$\sum_{n=2}^\infty \frac{F_n}{F_{n-1} F_{n+1}} = 1$$   Do you even use the formula $n(n+1)/2$?   I tried using $F_{n-1} = F_n - F_{n-2}$ and $F_{n+1} = F_n + F_{n-1}$ and then trying to figure out how to substitute those, but I've been going around in circles. How would you solve this? What tips can you give me to solve this? Is there any website that teaches this? I tried searching google and youtube but none were helpful.   2) Let $s_x = \tan(2/x)$. Where does this become nonincreasing? While using monotonic limits to establish there is a limit.   I was guessing you need to use upperbound and lowerbound but doesn't it go to infinity?
122653
Let $x, \ y, \ p$ be any real numbers with $x&gt;0$, $y&gt;0$, and $p&gt;1$.   The question is about (most probably) an elementary inequality:  Is it always true that  $x^p+y^p\leq (x+y)^p$ ?  Note that if $p$ is any positive integer, then the above inequality is obviously correct. What about if the number $p \ (\text{with} \ p&gt;1)$ is any non-integer real number?  I guess that (by my intuition) the answer should be positive. But how can we proceed to prove this inequality?
46332
OK guys I have this problem:  For $x,y,p,q&gt;0$ and $ \frac {1} {p} + \frac {1}{q}=1 $ prove that $ xy \leq\frac{x^p}{p} + \frac{y^q}{q}$  It says I should use Jensen's inequality, but I can't figure out how to apply it in this case. Any ideas about the solution?
23115
$(i)$ $a&gt;0,b&gt;0,p&gt;0,q&gt;0$   $(ii)$ $\frac{1}{p}+\frac{1}{q}=1$     How can we prove it?   ${a}{b}≤\frac{a^p}{p}+\frac{b^q}{q}$
38666
$\sum\limits_{n=1}^{\infty}\frac{1}{1+z^n}$, $|z|&gt;1$.  There are two facts that my professor uses that I am confused about.  The first is: $|1+z^n| \geq ||z|^n-1|$, I believe this is true for any $|z|$.  The other is: $\frac{1}{|z|^n-1} \leq \frac{2}{|z|^n}$, I believe this is also true for any $|z|$.  Can anyone prove these statements for me?
110909
Show that $\lim_{p \to 0} \mathbb {E} [(X^p-1)/p ] = \mathbb{E} [\log X]$   I am trying to show this equality by using the Dominated Convergence Theorem. Here $X$ is a non-negative random variable and $\mathbb {E}[X^p] &lt; \infty$ for some $p &gt;0$. I already found that $\lim_{p\to 0}\frac{X^p-1}{p} = \log X$. However, I can't find an uniform bound of $\frac{X^p-1}{p}$ for all $p$. Can anyone give me some hint on how to manipulate the sequence $\left\{\frac{X^p-1}{p}\right\}_{p&gt;0}$
1567
On , one can see that   $$\|fg\|_r\leq \|f\|_p\|g\|_q,$$ when $$\frac{1}{r} = \frac{1}{p} + \frac{1}{q}, \ p,q\in(0,\infty].$$  I have two questions,   Is the statement true? If so, is it stated in some references? I need it for my thesis.
76185
How can I prove that this function is bounded using the given 2 hints? $$f(x) = \frac{x}{x^2 + 1},$$  Hints:   $(x + 1)^2 \geq 0.$ $(x - 1)^2 \geq 0.$   thanks!
79323
I am trying to show that, for all $x&gt;0:$  $$\frac{1}{\sqrt{x}}\geq \frac{\ln x}{x-1}$$  This inequality is closer than I expected. I have tried exponentiating, power series, and have achieved nothing. I would really appreciate some help. Below is a graph of the two functions for small $x:$
119835
I'm trying to prove the convergence of $$ \sum_{n=1}^{\infty}\frac{1}{n^\alpha}$$ with $\alpha &gt; 1$.   For $\alpha \geq 2$ I can use the comparison test ($\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges) so I'm missing $2&gt;\alpha&gt;1$ and I'm pretty much out of ideas.  If you could offer some advice I would very much appreciate it. Thanks in advance
114720
I got stuck on this inequality for a day. If $p$ is positive integer, then the problem becomes too easy, but I can't find how we deal with the general case when p can be any positive number. Can someone give me a hint? I really appreciate     Prove that $|x^p - y^p| \le p|x-y|(x^{p-1} + y^{p-1})$ provided that $1 \le p \lt \infty$ and $x, y \ge 0$
38720
Let $f_n(x) = x^n(1-x)$ and $\sum (-1)^nf_n(x)$.  I showed that this series point wise.  Case1) $x=1$ $$ f_n(1)=0 \, , \sum{(-1)^nf_n(1)} = 0 $$ Case2) $x\neq 1$ $$|f_n(x)| \leq |x|^n$$ since $$\sum |x|^n$$ converges, By abosultely convergese test $\sum (-1)f_n(x)$ converges.  I want to show that this series uniformly converges on $[0,1]$.
106188
Let $f:[0,1]\rightarrow\mathbb{R}$ be a continuous function such that for all $a,b\in [0,1]$ with $a&lt;b$ $$f\left(\frac{a+b}{2}\right)\leq\frac{1}{b-a}\int_a^b f(x)\,dx.$$ How to prove that $f$ is a convex function? Thank you for your help.
69117
The more detailed question.    I'm kinda having some trouble starting out with answering this question. My initial approach would be to $g(x)= x^2$ since that is a convex function and find the expected values of $(T-P)^2$ and $(S-P)^2$.  However I fear that that would be too specific since the question would like to prove for any convex function. Is there a way to prove this more generally?
20622
I can't seem to work out the inequality $(\sum |x_n|^q)^{1/q} \leq (\sum |x_n|^p)^{1/p}$ for $p \leq q$ (which I'm assuming is the way to go about it).