Prove that $\lim\limits_{n \to \infty}\frac1n \sum\limits_{k=1}^nn^{1/k}=2$ Prove that $$\lim\limits_{n \to \infty}\frac1n \sum\limits_{k=1}^nn^{1/k}=2$$   How to estimate the sum? I tried to use Stolz's theorem but it is still a similar summation.
Methods to find $\lim\limits_{n\to\infty}\frac1n\sum\limits_{k=1}^nn^{1/k} $ What would you suggest here?  $$\lim_{n\to\infty} \frac{1}{n} \sum_{k=1}^{n} n^{1/k} $$
How does matter know to obey physical laws? First of all, you may say that, this is a "why" question and in physics we don't ask such a question and instead we ask about "how". However, I want to discuss this with you.  We model the behavior of matter by physical laws, for example by quantum mechanics laws. I want to know how the matter, say electron, knows that it must obey such a law. What forces the electron to obey quantum mechanics laws?  If you consider a simulation of the universe in a computer, in the view of someone  simulated in that computer the physical laws are just there. They may not know what keeps  in check the simulated matter to obey physical laws, but in our perspective those simulated physical laws are there because we implemented them as zero and ones in a hard disk. If we wanted, we could change those physical laws and in the eye of the simulated person everything would be still fine.  Is there a scientific attempt to discuss this issue? so that I can read about it. Is there a term that means "a person who enjoys learning"? This term might be used to describe someone who:   Is a self-motivated learner. Is curious, wants to understand many things.   I understand the term "philosopher" might be a good fit, in terms of its root words, however, the general population has an inconsistent understanding of this term, so I am looking for a more precise term. Is there a Ubuntu sanity check? Is there a standard procedure to 'unit test' an Ubuntu system against the deb packages?  Assume, a system has ubuntu-desktop installed according to the package manager. Can I find out, whether the file system has all the files in ubuntu-desktop and dependencies?   related question: Why is the inclusion of the tensor product of the duals into the dual of the tensor product not an isomorphism? Let $V$ and $W$ be vector spaces (say over the reals). There is a linear injection $V^* \otimes W^* \to (V \otimes W)^*$ which sends $\sum_i f_i \otimes g_i \in V^* \otimes W^*$ to the unique functional in $(V \otimes W)^*$ sending $v \otimes w \mapsto \sum_i f_i(v) \cdot g_i(w)$ for all $(v,w) \in V \times W$. In the finite dimensional case it is easy to see this is an isomorphism by comparing dimensions on both sides. I'm aware that this is not an isomorphism in the infinite-dimensional case (see Relation with the dual space in the  on tensor products) but I must admit that I'm not sure why. Cardinal arithmetic does not seem help here and, even if it did, I would be much happier to see an explicit example of a functional in $(V \otimes W)^*$ outside the range of this map. I'm having trouble cooking one up myself.  Added: Let me elaborate on my comment about cardinal arithmetic. Suppose that $X$ and $Y$ are infinite dimensional vector spaces over a field $k$. I'm confident that $\dim (X \otimes Y) = \dim X \cdot \dim Y$ holds. It is clear that $|X^*| = |k|^{\dim X}$ since we may identify a functional on $X$ with a function from a basis for $X$ to $k$. Also I think I've convinced myself that if $\dim X \geq |k|$ then in fact $|X| = \dim X$ ie. the cardinality and dimension of a vector space agree when the dimension is larger than the cardinality of the ground field. Consequently, assuming say that $|k| \leq \dim X \leq \dim Y$ it seems we have $$ \dim(X \otimes Y)^* = |k|^{ \dim X \cdot \dim Y} = |k|^{ \dim Y} = \dim Y^* = \dim X^* \cdot \dim Y^* = \dim (X^* \otimes Y^*)$$  which implies that in fact $(X \otimes Y)^*$ and $X^* \otimes Y^*$ are isomorphic but, rather frustratingly, the obvious map does not do the job. Did I make a mistake here? If not, is it generally true (ie without making assumptions about $|k|$) that there exists some isomorphism $(X \otimes Y)^* \to X^* \otimes Y^*$? Why the definition of limit have a strict inequality rather than not? My question is why the definition of limit of sequence is for $\epsilon &gt;0$, there exist $N$ st for $n\ge N$ that $|x_n-L|&lt;\epsilon$ instead of $|x_n-L|\le \epsilon$ Has anyone managed to get the Citrix receiver client (icaclient) working in Ubuntu? The latest beta update made the app crash a lot:   when I press 'Add comment'. after deleting a comment. after up/down/unvoting a post (happened when I upvoted an answer) after editing a post (but the edit is applied).   Spec: Nexus 5 on Lollipop 5.1.1, SE App v1.0.66 (beta) I know this subject was discussed for many times, but I still don't get it..  I've read in a book that we can use "I've got" instead of "I have" for possesion and it gave some examples,      I've got an English book.   I've got a red car.   So I understand that it can be used only for possession of physical objects, but I saw it used in sentences like "I've got something to say", is it correct? I hope it's not a stupid question..Thank you very much! How can I get a reference to a method? I have to create a report on some student completions. The students each belong to one client. Here are the tables (simplified for this question).  CREATE TABLE  `clients` (   `clientId` int(10) unsigned NOT NULL auto_increment,   `clientName` varchar(100) NOT NULL default '',   `courseNames` varchar(255) NOT NULL default '' )   The courseNames field holds a comma-delimited string of course names, eg "AB01,AB02,AB03"  CREATE TABLE  `clientenrols` (   `clientEnrolId` int(10) unsigned NOT NULL auto_increment,   `studentId` int(10) unsigned NOT NULL default '0',   `courseId` tinyint(3) unsigned NOT NULL default '0' )   The courseId field here is the index of the course name in the clients.courseNames field. So, if the client's courseNames are "AB01,AB02,AB03", and the courseId of the enrolment is 2, then the student is in AB03.  Is there a way that I can do a single select on these tables that includes the course name? Keep in mind that there will be students from different clients (and hence have different course names, not all of which are sequential,eg: "NW01,NW03")  Basically, if I could split that field and return a single element from the resulting array, that would be what I'm looking for. Here's what I mean in magical pseudocode:  SELECT e.`studentId`, SPLIT(",", c.`courseNames`)[e.`courseId`] FROM ... Loading a .X model in DirectX 9 Why isn't The Flash unstoppable? I've recently gotten into The Flash TV show, and so I've been reading about his powers. What I don't understand is why his typical rogues pose a threat - it seems like there should be very little limit to his powers.  He can go back in time, so if he loses a battle or the bad guys do something, he can go back and preempt it. In the new 52, he can all but predict the immediate future, so he can see bad things that are going to happen and stop them. He can phase matter through matter, or let matter pass through him. This means he is all but immune to weapons fire.  I haven't read enough of the old comics so I don't the know challenges he's had. For a guy who can't be attacked, can time-travel and can race around the globe several times in a minute, I can't imagine there is much that could pose a threat. What are the “spam” and “rude or abusive” (offensive) flags, and how do they work? How to tell whether a field is required or not in Apex by describe fields Java: Cannot find symbol? In the Stack Overflow podcast #36 (), this opinion was expressed: Once you understand how easy it is to set up a state machine, you’ll never try to use a regular expression inappropriately ever again. I've done a bunch of searching.  I've found some academic papers and other complicated examples, but I'd like to find a simple example that would help me understand this process.  I use a lot of regular expressions, and I'd like to make sure I never use one &quot;inappropriately&quot; ever again. The range of $T^*$ is the orthogonal complement of $\ker(T)$ How can I prove that, if $V$ is a finite-dimensional vector space with inner product and $T$ a linear operator in $V$, then the range of $T^*$ is the orthogonal complement of the null space of $T$?  I know what I must do (for a $v$ in the range of $T^*$, I have to show that $v\perp w$ for every $w$ in $\ker(T)$ and then do the opposite), but I don't know how to show that this inner product is zero. is it possible to trigger an email using microsoft flow when a date in a field of a sharepoint list item is nearby, I want to build a flow to trigger an email when a date in one of the fields is nearing. I have a field named Payment due date. is it possible to trigger an email 2 days prior to the Payment due date. So that I don't miss the payment due date and payment is not delayed. Any kind of assistance is appreciated. Thank you very much in advance! Follow a pipe using less? Can less follow (by pressing F) a piped input (similarly to a file)? For a file that is being written to, the command  less &lt;file&gt;   will follow the file when pressing F.  But if I have a command that pipes output directly into less, like this  command | less   pressing F will do nothing.  So it looks like pipes cannot be followed like files can? Or maybe it has to do with command also writing to STDERR? The effect I'm trying to achieve is always see the latest output of the command: just like keeping PageDown pressed!  A related remark holds for G (go to end): when piping directly to less, it won't work. of a monic polynomial $f \in \mathbb F\left[x\right]$ in $1$ variable $x$ over a field $\mathbb F$ plays an important role in understanding the structure of finite dimensional $\mathbb F[x]$-modules.  It is an important fact that the characteristic polynomial and the minimal polynomial of $C(f)$ are both equal to $f$. This can be seen quite easily by induction on the degree of $f$.  Does anyone know a different proof of this fact? I would love to see a graph theoretic proof or a non inductive algebraic proof, but I would be happy with anything that makes it seem like more than a coincidence!