I have tried to use induction, but after I assume that P(n) is true, I can't go further to prove that P(n+1) is true as well. I also have tried to find an intermediate inequality, but I can't figure out which inequality I should start from. Something that seemed to be useful was taking P(n) and multiplying it by $(1+\frac{1}{(n+1)^3})$, therefore I have come to this $(1+ \frac{1}{1^3})(1+\frac{1}{2^3})...(1+\frac{1}{n^3})&lt;3 | \times(1+\frac{1}{(n+1)^3})$ $(1+ \frac{1}{1^3})(1+\frac{1}{2^3})...(1+\frac{1}{n^3})(1+\frac{1}{(n+1)^3})&lt;3(1+\frac{1}{(n+1)^3})$ but, as anyone could imagine, I came to contradiction because I've tried to prove that $3(1+\frac{1}{(n+1)^3})&lt;3$ which is false. Any help it would be useful.
Prove that $\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)&lt;3$   for all positive integers $n$   This problem is copied from Math Olympiad Treasures by Titu Andreescu and Bogdan Enescu.They start by stating that induction wouldn't directly work here since the right hand side stays constant while the left increases.They get rid of this problem by strengthening the hypothesis.$$\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)\le3-\dfrac{1}{n}$$ and then proceed by induction.  The problem is that I can't find a motivation for the above change.I mean,we could have subtracted a lot of things from the RHS but what should nudge us to try $\dfrac{1}{n}$?The rest of the proof is quite standard,but I can't see how I am supposed to have thought of it.Is it just experience?Or is it a standard technique?A little guidance and motivation will be appreciated.
Grant admin rights on an application I'm want to grant a user administrative rights to one application on Windows 7. I don't want them to be able to have full admin rights, but admin rights are required for one of the applications they need to use.   Is there a way to set up a batch script or something where the user won't know it's being run as admin?  Or is there another way?  How can I accomplish this? Given: $\operatorname{gcf}(x,36)=9$ and $\operatorname{lcm}(x,36)=108$. Find $x$.   p.s I tried $54$ but although the $\operatorname{lcm}$ of $54$ and $36$ is $108$ the GCF is $18$ and it has to be $9$. Help me please. How can I escape special HTML characters in JSP? Do I choose the target or the ally for the Help action? In SF1, Stop refreshing the embedded vf page on a layout Below is my VF page which is embedded on a layout. In SF1 mobile app, it's getting refreshed whenever we touch on it. How can I avoid refreshing the VF page while touching it on the screen? In Desktop it's working fine. Please provide your inputs. &lt;apex:page standardController=&quot;PremiseObj__c&quot; lightningStylesheets=&quot;true&quot;&gt; &lt;apex:includeLightning /&gt; &lt;apex:slds /&gt; &lt;div id=&quot;premiseServDetails&quot;&gt; &lt;/div&gt; &lt;script&gt; window.onload = function (){     $Lightning.use(&quot;c:PremiseObjUpdateApp&quot;, function() {         $Lightning.createComponent(&quot;c:PremiseObjUpdatecmp&quot;,                                    {                                         recordId : &quot;{!Premiserecord__c.Id}&quot;                                     },                                    &quot;premiseServDetails&quot;,                                    function(cmp,event) {                                        setTimeout(function() {                                            //document.getElementById('loading').style.display = 'none';                                        }, 3000);                                    });     }); } &lt;/script&gt; &lt;/apex:page&gt; How do I use the Chinese remainder theorem to find all the square roots of 11 in $\mathbb Z_{35}$? Drupal SA-CORE-2014-005 - How to tell if my server / sites were compromised? I just updated all my sites using the patch method of resolving the Drupal SA-CORE-2014-005 exploit. I just read reports that just yesterday there is someone from a Russian IP infiltrating drupal sites.     My main concerns are now:   How do I tell if my sites have been comprised? What should I search for in my apache access logs to detect if my site was a victim or not? So far what are these hackers doing to comprised sites? Describe the units in $\mathbb{Z}[\sqrt{-d}]$ throwing exceptions out of a destructor Most people say never throw an exception out of a destructor - doing so results in undefined behavior. Stroustrup makes the point that "the vector destructor explicitly invokes the destructor for every element. This implies that if an element destructor throws, the vector destruction fails... There is really no good way to protect against exceptions thrown from destructors, so the library makes no guarantees if an element destructor throws" (from Appendix E3.2).   seems to say otherwise - that throwing destructors are more or less okay.  So my question is this - if throwing from a destructor results in undefined behavior, how do you handle errors that occur during a destructor?  If an error occurs during a cleanup operation, do you just ignore it? If it is an error that can potentially be handled up the stack but not right in the destructor, doesn't it make sense to throw an exception out of the destructor?  Obviously these kinds of errors are rare, but possible. How to clone all remote branches in Git? I have a master and a development branch, both pushed to . I've cloned, pulled, and fetched, but I remain unable to get anything other than the master branch back.  I'm sure I'm missing something obvious, but I have read the manual and I'm getting no joy at all. Tensor product of a sheaf of module and a residue field Let $S$ be a scheme over a finite field $\mathbb{F}_q$ and $\mathcal{L}$ an invertible sheaf on $S$.   In Pink's article (, see section 1.4 on page 13), he states that:  For any section $\ell\in \mathcal{L}(S)$ and any point $s\in S$ we let $\ell(s)\in \mathcal{L}\otimes_{\mathcal{O}_S}k(s)$ denote the value of $\ell$ over the residue field $k(s)$ of $s$.  My question is: what is the definition of $\mathcal{L}\otimes_{\mathcal{O}_S}k(s)$?  We know we can define the tensor product of two $A$-modules or of two $\mathcal{O}_X$-modules, but what is the tensor product of an $\mathcal{O}_X$-module and a field? Why is uniform prior on log(x) equal to 1/x prior on x? I'm trying to understand Jeffreys prior. One application is for 'scale' variables like the standard deviation $\sigma$ (or its square, the variance $\sigma^2$) of Gaussian distributions. It is often said that using a uniform prior over $\sigma$ is not really non-informative and instead one should either:   Use instead $\ln \sigma$ as the free parameter, with a uniform prior (this is often called a log-uniform prior) Or keep using $\sigma$ as the free parameter but use $1/\sigma$ as the prior (which is not uniform).   Why are the above two methods/priors equivalent? I feel it has something to do with the fact that the derivative of ln $\sigma$ is $1/\sigma$ but I can't take the next step.   Also, why does this even matter, in simple language with minimal jargon? I see all these complicated explanations online involving the Fisher information matrix but in the end all I see is that the above log-uniform or $1/\sigma$ priors preferentially weight lower values of $\sigma$ more highly. Why? If possible, a simple analytic example or python snippet would be very helpful. Alert on edit after a downvote Let $P$ denote the quotient space obtained by the action of $\mathbb{Z}\backslash2\mathbb{Z}$ by the antipodal map $z\mapsto\frac{1}{z}$ on the riemann sphere $\hat{\mathbb{C}}$ (identified here with $\mathbb{C}\cup\left\{\infty\right\}$). I identify $P$ with the set:  $\left\{ z\in\mathbb{C}:\left|z\right|&lt;1\right\} \cup\left\{ e^{it}:0\leq t\leq\pi\right\} $   that is to say, I use elements of the above set as the representatives for the equivalence classes in $P$. I am looking for formula for a function $f:P\times P\rightarrow[0,\infty)$ such that $f$ is a metric on $P$. Specifically, I would like a formula for $f$ that I can evaluate by plugging in representative elements in the above set (or something like that, more or less).  Thanks in advance.  Edit: forgive me for sounding desperate, but I cannot make due with an explanation of how to obtain such a formula. I want the formula.   An analogy for you if I have yet to make myself clear: suppose I was asking for the area of a square with side length $s$. The answers I have received so far for my question are akin to saying "multiply $s$ by itself" or "use the area formula for a square". The answer I am looking for is akin to saying "$s^{2}$". I need the formula. And please, no expressions with differentials, nor matrices, or any of that. I need to know how to compute the metric by using the complex numbers in the indicated set that I have identified with $P$. Boundary of Ball of radius R has zero measure Reason for LCM of all numbers from 1 .. n equals roughly $e^n$ I computed the LCM for all natural numbers from 1 up to a limit $n$ and plotted the result over $n$.  Due to the fast-raising numbers, I plotted the logarithm of the result and was surprised to find a (more or less) identity curve ($x=y$).  In other words, $LCM(1, 2, 3, ..., n)$ appears to be roughly the value $e^n$.  Is a there a simple explanation on why this is so?  $LCM(a, b, c, …)$ shall be defined as the least common multiple of all arguments $a, b, c, …$ Finding the number of zeros of $f(z) = z^n$ if $|f(z)| < 1 $ for all $z$ with $|z|=1$. Suppose $f: \overline{\mathbb{D}} \to \mathbb{C}$ is continuous, analytic in $\mathbb{D}$ and satisfies $|f(z)|&lt;1$ for $|z|=1$. Find the number of solutions to the equation $f(z) = z^n$ where $n$ is a positive integer. ($\mathbb{D}$ is the unit disk).  Some of my own attempts: Using the Maximum/Minimum Principle, we can split up the the problem in $2$ parts: Suppose $f$ is constant. It then holds that $z^n = c$ has exactly $n$ zeros because of the fundamental theorem of algebra.  Suppose $f$ is not constant. It holds that $f$ has at least one zero  in $\mathbb{D}$. It of course also holds that $z^n$ has a zero of multiplicity $n$ at the origin. I think I'm supposed to hit with stuff like Rouché's theorem, the Argument Principle or Schwartz lemma, but nothing seems to fit.   Thanks in advance. Signed to unsigned conversion in C - is it always safe? Proof for triangle inequality for vectors Generally,the length of the sum of two vectors is not equal to the sum of their lengths. To see this consider the vectors $u$ and $v$ as shown below.  By considering $u$ and $v$ as two sides of a triangle, we can see that the lengths of the third side is $\| u + v \|$ and we have $\| u + v \| \leq \|u\| + \|v\|$.  Under what circumstance equality occurs and how can one prove that? How do I set environment variables from Java? How do I set environment variables from Java?  I see that I can do this for subprocesses using . I have several subprocesses to start, though, so I'd rather modify the current process's environment and let the subprocesses inherit it.  There's a System.getenv(String) for getting a single environment variable. I can also get a Map of the complete set of environment variables with System.getenv(). But, calling put() on that Map throws an UnsupportedOperationException -- apparently they mean for the environment to be read only. And, there's no System.setenv().  So, is there any way to set environment variables in the currently running process?  If so, how?  If not, what's the rationale?  (Is it because this is Java and therefore I shouldn't be doing evil nonportable obsolete things like touching my environment?)  And if not, any good suggestions for managing the environment variable changes that I'm going to need to be feeding to several subprocesses?