Let $a_1,...,a_n$ be positive numbers. Show that $(a_1+...+a_n)(\frac{1}{a_1}+...+\frac{1}{a_n})\ge n^2$. Just not really sure how to approach this problem, any tips/solutions would be greatly appreciated.
If $x_1,...,x_n$ are positive real numbers and if $y_k=1/x_k$, prove that $$\left(\sum^n_{k=1}x_k\right)\left(\sum^n_{k=1}y_k\right)\geq n^2.$$   I've been learning induction, and I've come across this problem that I really can't even break down and begin to think about. I've been told it has something to do with Cauchy-Schwarz, but I cannot figure out how to apply it. I would appreciate help figuring out how to go about and formulate this proof. Thanks!
show that $$\lim_{n\to \infty}\sin{(n^m)}$$  divergent,where $m\in N^{+}$  I have kown $$\lim_{n\to\infty}\sin{n^2}$$  to be divergent and dense in $[-1,1]$. This is very  famous problem,the problem is first is post $AMM$( The American Mathematical Monthly,1970-1975) problem,and after some years  $AMM$ post $\sin{(n^{14})}$ in $[-1,1]$ was dense? and this is open problem.so I Guess  $$\sin{(n^m)}$$ is dense in $[-1,1]$. How does the two slit experiment work? Whilst going through 'A brief history of time', I faced difficulty in understanding the . How can an individual electron cause fringes on screen? I was unable to understand it? Please explain it simply. Is there anybody who can help me show the following?      $$ \sum_{x=0}^{n-1} \cos\left(k +x{2\pi\over n}\right) =0  \qquad\hbox{and}\qquad  \sum_{x=0}^{n-1} \sin\left(k +x{2\pi\over n}\right) =0 $$   I can only prove this claim for the sine expression when $k=0$, by grouping the output values such that they all cancel out (for both $n$ even and odd). I have thought about using complex numbers in exponential form to prove it, but I can't do it ... :P   I would really appreciate an algerbraic proof. How can I correct a reviewer's opinion? I have my own AppleID ("Bob") but I don't use this email addy "Bob" anymore. I want to add my current, daily used email addy ("Jim") to my ID, or better yet, make it my only AppleID and delete "Bob". BUT... I already used "Jim" for an AppleID for an older friend who had no email address, so set it up for him on mine, "Jim".  My older friend now has an email address ("Ivan") so can I   add "Ivan" to his AppleID, make it the primary email address, and then delete "Jim" from it? Or... delete his AppleID and create a new one for him "Ivan" knowing he would lose his minor purchases   And most importantly, this would all be pointless unless: - will deleting/removing "Jim" as linked to my friend's AppleID then make "Jim" available for me and no longer registered to another account? This is one of the possible ways I come out:  struct RetrieveKey {     template &lt;typename T&gt;     typename T::first_type operator()(T keyValuePair) const     {         return keyValuePair.first;     } };  map&lt;int, int&gt; m; vector&lt;int&gt; keys;  // Retrieve all keys transform(m.begin(), m.end(), back_inserter(keys), RetrieveKey());  // Dump all keys copy(keys.begin(), keys.end(), ostream_iterator&lt;int&gt;(cout, "\n"));   Of course, we can also retrieve all values from the map by defining another functor RetrieveValues.  Is there any other way to achieve this easily? (I'm always wondering why std::map does not include a member function for us to do so.) Why was this very low quality flag declined? Question (since closed and deleted): . I know about the  of VLQ flags, and I can't fathom why anyone would think that question wouldn't be a candidate for removal.  Question text, for posterity:     Hey I want a Cyrus beck line algorithm program in c or c++. I have   searched a lot but hadn't got anything over Web..!!      If anyone of you have Program please give it to me and help me out.      Thank you in advance :)   I find  not compelling because it requires me to accept the premise that this question was not "unlikely to be salvageable". If I said, &quot;I will run, not if it is snowing, if I my shoes are comfortable.&quot; Does the &quot;if my shoes are comfortable&quot; modify the &quot;I will run&quot; clause? I agree the sentence would be clearer with the addition of &quot;and&quot; in place of the comma. However, without the &quot;and&quot; is it still clear that the &quot;if my shoes are comfortable&quot; clause modifies the &quot;I will run&quot; clause? How to edit files in a terminal with nano? In R, what does qqnorm actually do? qqnorm(x) plot(qnorm(seq(80)/81),sort(x))   Finding that the plots produced by the commands above are slightly different from each other, I tried this:  qqnorm(qnorm(seq(80)/81))   I got a slightly less than perfect line.  I'd have tried regressing the result of qnorm(seq(80)/81) on the variable that was plotted on the $x$-axis and plotting residuals against the predictor, expecting to see some graceful curve, but for the fact that I don't know what to use as the predictor.  Possibly such a residual plot would reveal more than just the graceful S-shaped thing I'd anticipate.  So my question is this: if the thing on the $x$-axis in the plot produced by qqnorm is not what I get from qnorm(seq(80)/81) (and what I did shows that indeed it is not), then what is it? I'm trying to prove that every finite Boolean ring, $R$, is isomorphic to a finite number of copies of $\mathbb{Z} / 2 \mathbb{Z}$:  \begin{align} R \cong \mathbb{Z} / 2 \mathbb{Z} \; \times \cdots \; \times \; \mathbb{Z} / 2 \mathbb{Z} \end{align}  I'm aware that there are answers to this question on this website. However, almost all of the answers seem to use facts about modules and vector spaces defined over fields. This question is posed in section 7.6 in Dummit and Foote's Abstract Algebra. This section is part of the chapters which introduces the basics of ring theory (rings, homomorphisms, ideals, Chinese Remainder Theorem etc.).  I'm not even sure how to being answering this question. I know that if $R$ were an integral domain, then we would have that $R \cong \mathbb{Z} / 2 \mathbb{Z} $.   Please only give hints, considering that that the statement has to be proved using the material only included up till chapters 9 in the textbook. The SPListItem being updated was not retrieved with all taxonomy fields I want to turn a photo in a engraving style picture, like  using GIMP and, if possible, then make an SVG out of it with Inkscape.  Can you recommend any filters/tools/plugins/tutorials for this? I tried to install OracleJDK 8 using   sudo add-apt-repository ppa:webupd8team/java   But it's not working.   I searched the problem and found that      "The Oracle JDK License has changed for releases starting April 16, 2019." And  "Oracle Java (JDK) 8 Installer PPA (DISCONTINUED)"   So I installed the OpenJDK8 .   But why oracle cancelled the license of oracle 8 installer? I want to prove that $\lim_{(x,y)\to (0,0)} \frac{\sin(x+y)}{x+y} = 1$  Is it sufficient to say that if $u = x+y$ then as $(x,y) \rightarrow (0,0)$ then $u \rightarrow 0$ and then evaluate $\lim_{u\to 0} \frac{\sin u}{u}$ which is 1 by L'Hôpital's rule?  I feel that this is somehow "cheating" and I would really like to prove it in a more rigorous way, i.e., by using an $\epsilon$ - $\delta$ argument.  I have tried for a long time to use the one variable version and prove that if $0 \lt \sqrt{x^2 + y^2} \lt \delta$ then somehow $0 \lt |x+y| \lt \delta$ and it would follow from the one variable case that $| \frac{\sin(x+y)}{x+y} - 1| \lt \epsilon$. However, because $|x+y| = 0$ when $y = -x$ I am not getting anywhere with this.  Any input on this would be helpful :) Notepad++ incrementally replace Why are my fields initialized to null or to the default value of zero when I've declared and initialized them in my class' constructor? How to use unicode characters in Windows command line? How to convert light activated switch Can I declare a duplicate page via javascript for google? Most sources of documentation state that one can declare a webpage a duplicate of another by applying this tag on the duplicate page:  &lt;link rel="canonical" href="(absolute URL to original page)"&gt;   I could generate the above tag via the script below:  &lt;script&gt; var l=document.createElement('Link'); l.rel='canonical';l.href='(absolute URL to original page)'; document.getElementsByTagName('head')[0].appendChild(l); &lt;/script&gt;   My question is, would google be able to process this javascript and believe I'm declaring the page as a duplicate page? or would I be forced to use the link tag in the header?  I ask because I'm trying to reduce the unnecessary bytes required to be processed before the actual user readable content begins.