I have to prove this inequality.  $$\sqrt[n]{n} - 1 \lt \sqrt{\dfrac 2n} ~~\mbox{for}~~ n \gt 2$$  So far I reached this result:  $$n^{\dfrac {n+2}{n}} - n^{\dfrac {n+1}{n}} + n - n^{\dfrac {n+1}{n}} \lt 2$$
Prove that for any positive integer $n$,       $$n^{1/n}  &lt; 1 + \sqrt{\frac{2}{n}}.$$   This due to Victor Linis, Eureka, Vol. 2, No. 2, February 1976, p. 29.   Hint:      Use the binomial theorem.
Bjarne Stroustrup once mentioned in one of his bigthink interviews that one of the greatest problems most programmers have is not finding or using the right libraries.  On stackoverflow and programmers, I've observed a trend towards questions like "I want to do this, is there a [library | programming language] for that", or "I have these {n} [library | language] candidates for my project; which one would yield me the least amount of headache given these requirements".   Since questions like these aren't exactly programming questions in the sense of "this code should do this, but it does this, why?", they are often (though not always) being closed as "not constructive" because they do not have a definite answer (which is true, and according to the FAQ they would be off-topic on both stackoverflow and programmers).  Nonetheless, I think library questions are valid, and that it is helpful for any programmer to receive independent input from experienced programmers (at least it helps me a lot every time I browse through one of those closed questions, which I found through google). Does anyone agree that posting library choice questions should be explicitly allowed on either stackoverflow or programmers? sum of noncentral Chi-square random variables I need to find the distribution of the random variable  $$Y=\sum_{i=1}^{n}(X_i)^2$$ where $X_i\sim{\cal{N}}(\mu_i,\sigma^2_i)$ and all $X_i$s are independent. I know that it is possible to first find the product of all moment generating functions for $X_i$s, and then transform back to obtain $Y$'s distribution. However, I wonder whether there is a general form for $Y$ like the Gaussian case: we know the sum of independent Gaussian is still a Gaussian, and thus we only need to know the summed mean and summed variance.   How about all $\sigma^2_i=\sigma^2$? Will this condition make a general solution? Why do the $n \times n$ non-singular matrices form an "open" set? Why is the set of $n\times n$ real, non-singular matrices an  open subset of the set of all $n\times n$ real matrices? I don't quite understand what "open" means in this context.  Thank you. Distorted Cabin & Oily Sheen: What is going on in this paragraph? Proof that if continuous function on $[0, 2]$ and $f(0) = f(2)$, then for some $x: f(x) = f(x - 1)$ Markdown help page is broken (badly presented/formatted) on every site In the  I see:    Is that what the help is supposed to say and look like? Because it doesn't make sense to me.  As animuson points out in a comment, all of the highlighted example text is being shifted to the very left of each line. The entire page is broken. How to pin either a Shortcut or a Batch file to the new Windows 7, 8 and 10 Taskbar and start menu? We are having trouble adding our batch scripts to the Windows 7, 8 or 10 taskbar or start menu.  Our batchfiles take some arguments and just execute other application based on those arguments . To keep it simple, we created some shortcuts for our customers which they just need to click in order to get everything running.  The question here is: How can we pin those shortcuts or batch files to the taskbar or start menu?  (The customers aren't supposed to access the file system and that's why they can only see the taskbar) How to write a while loop with the C preprocessor? Minimal polynomial of $\sqrt{2}+\sqrt{3}$ over $\mathbb Q$ I had an example in the book given as follows:        Find the minimal polynomial of $\sqrt{2}+\sqrt{3}$ over $\mathbb Q$ .       Solution:  $~~~~~$$(\sqrt{2}+\sqrt{3})^2=5+2 \sqrt6$      $(\sqrt{2}+\sqrt{3})^4=49+20 \sqrt6$       Then $(\sqrt{2}+\sqrt{3})^4-10(\sqrt{2}+\sqrt{3})^2+1=0.$       Thus $a=\sqrt{2}+\sqrt{3}$ satisfies $f(x)=x^4-10x^2+1$ over $\mathbb Q$.       Let $p(x)$ be the minimal polynomial of $\sqrt{2}+\sqrt{3}$ over $\mathbb Q$.Then $(\sqrt{2}-\sqrt{3}),(-\sqrt{2}+\sqrt{3}),(-\sqrt{2}-\sqrt{3})$ are also roots of $p(x).$   So degree of $p(x)$ is atleast $4$.      But $f(a)=0 $ and $f(x) \in \mathbb Q[x] \implies p(x)$ divides $f(x)$ .      So $f(x)$ is minimal polynomial of $\sqrt{2}+\sqrt{3}.$   But I can't get the step  how are $(\sqrt{2}-\sqrt{3}),(-\sqrt{2}+\sqrt{3}),(-\sqrt{2}-\sqrt{3})$ also roots of $p(x).$   Kindly help with this. Are there any tools in Blender that take irregular polygons and make them regular (approximate a circle)? I'm specifically looking for a solution that works with preexisting polygons. Also, it would be nice if it keeps the overall position after making it regular.    Irregular Polygons    Regular Polygons Locking a file in Python find examples of two series $\sum a_n$ and $\sum b_n$ both of which diverge but for which $\sum \min(a_n, b_n)$ converges Find examples of two series $\sum a_n$ and $\sum b_n$ both of which diverge but for which $\sum \min(a_n, b_n)$ converges. To make it more challenging, produce examples where $a_n$ and $b_n$ are positive and decreasing.  Edit: This problem is taken verbatim from Exercise 2.7.11 on page 68 of Abbott's Understanding analysis. Is it possible to construct a multi-colored double-bladed lightsaber? I was just watching  video of some SW dueling at the 2015 World Fencing Championship, and wondered if it's actually possible in the SW universe to construct a multi-colored double-bladed lightsaber?   Are there any accounts of such a thing? If not, would it at least be theoretically possible to make one, or, are there obvious limitations? Probability that the bag contains all balls white given that two balls are white A bag contains $4$ balls .Two balls are drawn at random and are found to be white.  What is the probability that all the four balls in the bag are white?.  The options are $\large\frac{1}{5}\space ;\frac{2}{5} ;\frac{3}{5} ;\frac{4}{5}$  My work: It's given that two balls are white.So the rest of the balls could be    all white One of them white none of them white   So among the three above possibilities only one satisfies our conditions so the probability is $\frac{1}{3}$.But it not even the options given. What's wrong with my work? What is $A^-$ in topology? Hartshorne Algebraic Geometry Proposition 1.5 (page5) In a notherian topological space $X$, every nomempty closed subset $Y$ can be expressed as a finite union $Y=Y_1 \cup \cdots \cup Y_r$ of irreducible closed subsets $Y_i$. If we require that $Y_i \nsupseteq Y_j$ for $i \neq j$, then the $Y_i$ are uniquely determined. They are called the irreducible components of $Y$.    I understood the existence of such a representation. But I can't understand a sentence in  the uniqueness part, which I highlighted: Let $Y=Y_1 \cup \cdots \cup Y_r$ and $Y=Y&#39;_1 \cup \cdots \cup Y&#39;_s$ be two such representation. Then we can deduce that $Y_1=Y&#39;_1$ (the details are in the textbook). Now let $Z=(Y-Y_1)^-$, then $Z=Y_2 \cup \cdots \cup Y_r$ and also $Z=Y&#39;_2 \cup \cdots \cup Y&#39;_s$.  What is the operator $^-$? I understood it as a closure, is it right? But then how can I deduce that $Z$ is represented as $Z=Y_2 \cup \cdots \cup Y_r=Y&#39;_2 \cup \cdots \cup Y&#39;_s$? Infinite Product $\prod\limits_{k=1}^\infty\left({1-\frac{x^2}{k^2\pi^2}}\right)$ Difference between the use of "a" and "any" in these examples? I didn't have an idea.      I did not have any idea.   I am much more familiar with the second one. The first one sounds a bit odd to me, but sounds fine if I say:     I did not have even an idea.   But, the first one is also one of the widely used sentences by natives. What is the difference between the two? The difference between two periodic functions converges to zero, is this two functions identical? How to select multiple files with keyboard only in windows explorer Express partial derivatives of second order (and the Laplacian) in polar coordinates $z=f(x,y)$ where $x=rcosθ$ and $y=rsinθ$  Find $ \frac{\partial z}{\partial x}$ and $ \frac{\partial^2 z}{\partial x^2}$  I'm having big troubles with using chain rule, in particularly the second derivative. Spent 2 hours on this already. What I have...  $ \frac{\partial z}{\partial x}=\frac{\partial r}{\partial x}\frac{\partial z}{\partial r}+\frac{\partial z}{\partial θ}\frac{\partial θ}{\partial x}$. So I'm guessing that $\frac{\partial z}{\partial r}$ and $\frac{\partial z}{\partial θ}$ are unkown? And I will need to use them to differentiate again. Any help is really welcome. This is simply only part of a bigger question.   Full Question: