As very first task in our analysis I class we got the following task which gives (only) 1 of 16 points. I add the points in order to calculate the needed effort to solve this task...      So let $n,m\in\mathbb{N}$ and $a_n:=\sqrt[n]{b_1^n+...+b_m^n}$, with $b_i\geq 0$ and $i=(1,...,m)$ We have to prove if this sequence converges and determine the limit (by using $\epsilon$ definition).     After running a simple R simulation I found out, that this sequence seems to converge against m.     As I have learned so far, the first step is to determine an $N\in\mathbb{N}$ so that $\forall n\in\mathbb{N}, n\geq N : |a_n-m|&lt;\epsilon$, but while searching for that $N$ I began to struggle. $|a_n-m|=|\sqrt[n]{b_1^n+...+b_m^n}-m| &lt; \epsilon \Leftrightarrow ... \Leftrightarrow b_1^n+...b_m^n&lt;(\epsilon+m)^n$. After some research I found, that the LHS can be expressed by polynomials using . At this point I am afraid to be on the wrong track.     Can somebody give me a hint which respects the limitations (having only 1 lecture and it is the first task regarding sequences)?  The R-Code    m_max = 20 m=seq(1,m_max) sumOfPowers=function(n){sum(m^n)} n_max = 50 n=seq(1,n_max) temp=sapply(n,sumOfPowers) a_n=nthroot(temp,n) print(data.frame(n,temp,a_n))
Let $m\in \mathbb{N}$ and $c_1,c_2,\ldots,c_m \in \mathbb{R}_+$. Show that $$\lim_{n\rightarrow \infty} \sqrt[n]{c_1^n+c_2^n+\ldots+c_m^n} = \max\{c_1,c_2,\ldots,c_m\}$$  My attempt: Since $$\lim_{n\rightarrow \infty} \sqrt[n]{c_1^n+c_2^n+\ldots+c_m^n} \leq \lim_{n\rightarrow \infty}\sqrt[n]{\max\{c_1,c_2,\ldots,c_m\}} = \lim_{n\rightarrow \infty} \sqrt[n]{n}\sqrt[n]{\max\{c_1,c_2,\ldots,c_m\}}=\lim_{n \rightarrow \infty} \max\{\sqrt[n]{c_1^n},\sqrt[n]{c_2^n},\ldots,\sqrt[n]{c_m^n}\}=\lim_{n \rightarrow \infty}\max\{c_1,c_2,\ldots,c_m\}=\max\{c_1,c_2,\ldots,c_m\}$$  it follows that $\lim_{n\rightarrow \infty} \sqrt[n]{c_1^n+c_2^n+\ldots+c_m^n}$ is bounded, but I don't think it's monotonically decreasing, at least I can't prove this. Can anybody tell me whether the approach I have chosen is a good one, whether what I have done is correct and how to finish the proof?
Efficient online linear regression Wifi losing connection, weak signal, Intel 7260 adapter I yesterday upgraded from Windows to Ubuntu 15.04 but I have problem with my wireless adapter Intel Corporation Wireless 7260 (rev 73). It randomly loses internet connection. Restarting adapter always helps. (same it did with Windows, except that sometimes restarting adapter didn't help).   Additionally, I have weaker connection then I had with Windows.  Thank you! How do we see? Where do the photons disappear? I know that the light is reflected from a object to my eyes, but I don't understand exactly how. The photons appear from the light source and disappear in my eye! Can someone explain the phenomenon of where the photons go and do to allow us to see? How should an educational institute woo good faculty? What steps must an educational institution wanting to recruit good faculty take? There is a cycle in the academia:    Good doctorates from top universities invariably join other top universities as faculty members.  Good students aspire to study in top universities under excellent faculty members.   Suppose a new educational institution or department springs up and desires to set up a top-quality group, how does it break into this cycle? I suppose this could take some years, but is money-power the only thing that could be wielded by the institute to achieve its end? I am currently using Ubuntu 10.04. I know there is a 10.10 release, but can I upgrade directly to 11.04? Could you walk me through the steps please? Power Supply Capacity Formula Given that I am building a home computer from scratch and I have several components (MB, CPU, HDD, Video, Sound, etc) which have specific power requirements, is there a formula for me to calculate the size/capacity of the Power Supply I will need? I want to be able to expand the setup later on also... Sturm-Liouville Questions In thinking about Sturm-Liouville theory a bit I see I have no actual idea what is going on.  The first issue I have is that my book began with the statement that given  $$L[y] = a(x)y'' + b(x)y' + c(x)y = f(x)$$  the problem $L[y] \ = \ f$ can be re-cast in the form $L[y] \ = \ \lambda y$.  Now it could be a typo on their part but I see no justification for the way you can just do that!  More importantly though is the motivation for Sturm-Liouville theory in the first place. The story as I know it is as follows:  Given a linear second order ode   $$F(x,y,y',y'') = L[y] = a(x)y'' + b(x)y' + c(x)y = f(x)$$  it is an exact equation if it is derivable from a differential equation of one order lower, i.e.  $$F(x,y,y',y'') = \frac{d}{dx}g(x,y,y').$$  The equation is exact iff  $$a''(x) - b'(x) + c(x) = 0. $$  If $F$ is not exact it can be made exact on multiplication by a suitable integrating factor $\alpha(x)$.  This equation is exact iff  $$(\alpha(x)a(x))'' - (\alpha(x)b(x))' + \alpha(x)c(x) = 0 $$  If you expand this out you get the Adjoint operator  $$L^*[\alpha(x)] \ = \ (\alpha(x)a(x))'' \ - \ (\alpha(x)b(x))' \ + \ \alpha(x)c(x) \ = 0 $$  If you expand $L^*$ you see that we can satisfy $L \ = \ L^*$ if $a'(x) \ = \ b(x)$ &amp; $a''(x) \ = \ b'(x)$ which then turns $L[y]$ into something of the form  $$L[y] \ = \ \frac{d}{dx}[a(x)y'] \ + \ c(x)y \ = \ f(x).$$  Thus we seek an integratiing factor $\alpha(x)$ so that we can satisfy this &amp; the condition this will hold is that $\alpha(x) \ = \ \frac{1}{a(x)}e^{\int\frac{b(x)}{a(x)}dx}$  Then we're dealing with:  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ \alpha(x)c(x)y \ = \ \alpha(x)f(x)$$  But again, by what my book said they magically re-cast this problem as  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ \alpha(x)c(x)y \ = \ \lambda \alpha(x) y(x)$$  Then calling  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ ( \alpha(x)c(x)y \ - \ \lambda \alpha(x) )y(x) \ = \ 0$$  a Sturm-Liouville problem.  My question is, how can I make sense of everything I wrote above? How can I clean it up &amp; interpret it, like at one stage I thought we were turning our 2nd order ode into something so that it reduces to the derivative of a first order ode so we can easily find first integrals then the next moment we're pulling out eigenvalues &amp; finding full solutions - what's going on? I want to be able to look at $a(x)y'' \ + \ b(x)y' \ + \ c(x)y \ = \ f(x)$ &amp; know how &amp; why we're turning this into a Sturm-Liouville problem in a way that makes sense of exactness &amp; integrating factors, thanks for reading! Someone's car hit our metal bar gate, so one of the rods is bent as shown in the photo. What kind of a tool can I use to bend this back, so it's straight? Thank you! Can someone please help me thanks!  this is where I got the image from How can I view the contents of tar.gz file without extracting from the command-line? I want to see the contents (list of files and folders) of an archive, for example a  tar.gz file without extracting it.  Are there any methods for doing that? List all directories that do NOT have a file with a given file name inside How would I go into listing all directories that do not have a file with a given file name inside? e.g. given this tree  /   /a      README      file001      file002   /b      README      file001   /c      file003   I want to list the directories that do not have a file named README, in this case it would be directory /c. How would I do that? I can't think of any syntax using e.g. find. What do we lose passing from the reals to the complex numbers? As normed division algebras, when we go from the complex numbers to the quaternions, we lose commutativity. Moving on to the octonions, we lose associativity. Is there some analogous property that we lose moving from the reals to the complex numbers? If $X$ and $Y$ are independent $N(0,\sigma^2)$, then $X^2+Y^2$ and $X/Y$ are independent? If $X$ and $Y$ are independent, then $X^2+Y^2$ and $X/Y$ are independent? I was solving the problem for the case that $X$ and $Y$ are independent $N(0,\sigma^2)$.  So i found that $X^2+Y^2$ is negative exponential, $X/Y$ is Cauchy distribution. How can i prove that $X^2+Y^2$ and $X/Y$ are independent?  Also, what it will be if the case for general random variable $X$, $Y$ with  $X^2+Y^2$ and $X/Y$ are defined? Pdf of the square of a standard normal random variable What is the regular expression for a decimal with a precision of 2?  Valid examples:  123.12 2 56754 92929292929292.12 0.21 3.1   Invalid examples:  12.1232 2.23332 e666.76   The decimal point may be optional, and integers may also be included. The thing with these tags is, you can always (or almost always) use a specific product/technology/language tag which would render the company-name tag completely redundant (see Appendix C for sloppy non-proof). For example, what relevant information would  tag add to  already tagged with ? What would  tag add to  already tagged with ? Another way of looking at this: how much useful information do you lose if you don't tag a question with a company name? Note: I'm not talking about brand/company/organizations like , , ,  because those names also refer to websites/web services with which you can interact via API or from which you can extract data. Appendix C: The &quot;about&quot; test The relationship between a tag and a question can be expressed as such  This question is about [tag]  In case of ,  This question is about [microsoft]  Fair enough. But in reality people are using these company-name tags like this  This question is about something developed by or related to [microsoft]  But if that's the case, you might as well just go with  This question is about [something developed by or related to microsoft]  where  can be , , ... you get the idea. To find tangents to given circle from a point outside it Find the combined equation of two tangents drawn from $P(x_1,y_1)$ to the circle $x^2+y^2 = a^2$. Point $P$ lies outside the circle. Nested Quantifiers true or false I have a concern with nested quantifiers.  I have: $$  \forall x \exists y \forall z(x^2-y+z=0) $$ such that $$ x,y,z \in \Bbb Z^+$$   My first question, can it be read like this:  $$  \forall x \forall z \exists y(x^2-y+z=0)  $$  The way I did it, is I started off with $x=1, z=1 $   $$ 2-y = 0 $$ $$ y =2 $$  Is this a good approach? How to mount a remote (over SSH) directory to be available like a local directory (incl. in shell, not just in Gnome)? We have some new hardware in our office which runs its own customized Linux OS.   How do I go about figuring which distro it's based on?