Find the limit $$\lim_{n\to \infty}\{{(n!)}^{1/n}\}/n$$  I took exp log but getting answer as 1 but it should be 1/e. Required a nice approach.
I'm trying to find  $$\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}} .$$   I tried couple of methods: Stolz, Squeeze, D'Alambert  Thanks!  Edit: I can't use Stirling.
Prove $(x_1+.....+x_n)^2 \leq n(x_1^2 + .....+x_n^2)$ Prove that  $(x_1+\dots+x_n)^2 \leq n(x_1^2 + \dots+x_n^2)$  for all positive integers n and all real numbers $x_1,....,x_n$  I am attempting a proof by induction but wasn't sure if i need the Cauchy-Schwarz Inequality or perhaps another way other than induction to prove this.  Proof   $n=1$ true  assume true for $n=k$  Now for $n= k+1$  $(x_1 + \dots +x_k + x_{k+1})^2 \leq \dots$ Calculationg base-resistor between microcontroller and transistor I'm curious, when connecting my 2N2222 transistor's base to a PWM output pin on my Arduino, I know the Arduino can deliver a maximum of 40mA current. The 2N2222 only requires around 5 to 10 mA on it's base to switch the collector and emitter. Now, when calculating the base-resistor under "normal" conditions where I'm merely switching it directly from Vcc, the following applies:   However, when putting in a 500 ohm or 1K resistor in (1K for 5mA) directly it works great. But putting it in between the Arduino and the transistor, the transistor doesn't switch.  I suspect thus, that the resistance is too high of the base-resistor. Therefore, we may very well not need a base-resistor.  But, according to spec, the Arduino can deliver 40mA. So, am I correct in assuming the folling: 1) I should calculate the perceived resistance that the Arduino is giving, if it goes high to 5Volt and only gives 40mA, and then 2) Subtract that from the 500ohm resistor needed to make the transistor operate optimally.  Example:    So, I'm theorising that I have to then put in a 375 ohm resistor in to make the total resistance between the Arduino and the base resistor 500 ohm.  Somehow this doesn't feel right. Keep in mind, I'm not an electronic engineer, so I might have the cat by the tail here :P Show that if $f(z)$ is an entire function such that it takes real values on the lines $Im(z)=0$ and $Im(z)=a$ (for some $a \in \mathbb{R}_{\neq 0})$, then $f$ is periodic with period $2 i a$.    So here is my thought process: First of all by reflection principle $f(\bar{z})=\overline{f(z)}$ so if our function $f$ is given in the strip $0 \leq Im(z) \leq a$, then its values on $-a \leq Im(z) \leq a$ follow by reflection principle. Then by imposing $f(z)=f(z+2 ia)$ we can intuitively see that this will define an entire function by looking at how the strips $na \leq Im(z) \leq 2na$ seamlessly stack on top of each other. Is this logical? And how would one prove this analytically? How do I update Custom Label's value using apex? Which one would be a better option metadata api or tooling api? Please provide example How to hook into the CPT's title placeholder? How to perform Stream functions on an Iterable? How do I tell if matrices are similar? I have two $2\times 2$ matrices, $A$ and $B$, with the same determinant. I want to know if they are similar or not.  I solved this by using a matrix called $S$: $$\left(\begin{array}{cc} a&amp; b\\ c&amp; d \end{array}\right)$$ and its inverse in terms of $a$, $b$, $c$, and $d$, then showing that there was no solution to $A = SBS^{-1}$. That worked fine, but what will I do if I have $3\times 3$ or $9\times 9$ matrices? I can't possibly make system that complex and solve it.  How can I know if any two matrices represent the "same" linear transformation with different bases?  That is, how can I find $S$ that change of basis matrix?  I tried making $A$ and $B$ into linear transformations... but without the bases for the linear transformations I had no way of comparing them.  (I have read that similar matrices will have the same eigenvalues... and the same "trace" --but my class has not studied these yet. Also, it may be the case that some matrices with the same trace and eigenvalues are not similar so this will not solve my problem.)  I have one idea. Maybe if I look at the reduced col. and row echelon forms that will tell me something about the basis for the linear transformation? I'm not really certain how this would work though? Please help. What's the best way of accessing field in the enclosing class from the nested class? Do you need to erase the memory of *watching* a memory in a Pensieve? I'm having troubles with one of the problems in the book Introduction to Commutative Algebra by Atiyah and MacDonald. It's on page 11, and is the last part of the second question.  Given $R$ a commutative ring with unit. Let $R[x]$ be the ring of polynomials in an indeterminate $x$ with coefficients in $R$. We say that a polynomial $f = \sum\limits_{i=0}^n r_ix^i \in R[x]$ (with coefficients $r_0, r_1, \ldots, r_n$) is primitive if $\langle r_0,r_1,...,r_n\rangle = R$, i.e., the ideal generated by the coefficients of $f$ is $R$. Let $f$ and $g$ be two polynomials in $R[x]$. Prove that $fg$ is primitive iff $f$ and $g$ are both primitive.  The $\Rightarrow$ part is easy. Say, $f = \sum\limits_{i=0}^n r_ix^i$, $g = \sum\limits_{i=0}^m s_ix^i$; then $fg = \sum\limits_{i=0}^{m+n} c_ix^i$, where $c_k = \sum\limits_{i + j = k}r_is_j$. Since $fg$ is primitive, there exists a set of $\{\alpha_i\} \subset R$, such that $\sum\limits_{i=0}^{m+n} \alpha_ic_i = 1$, to prove $f$ is primitive, I just need to write all $c_i$'s in terms of $r_i$'s, and $s_j$'s, then group all $r_i$ accordingly, rearranging  it a little bit, and everything is perfectly done. And the proof of the primitivity of $g$ is basically the same. The $\Leftarrow$ part is just so difficult. Say $f = \sum\limits_{i=0}^n r_ix^i$, $g = \sum\limits_{i=0}^m s_ix^i$ are both primitive, then there exists $\{\alpha_i\}; \{\beta_i\} \subset R$, such that $\sum\limits_{i=0}^{n} \alpha_ir_i = 1$, and $\sum\limits_{i=0}^{m} \beta_is_i = 1$. At first, I thought of multiplying the two together, but it just didn't work. So, I'm stuck since I cannot see any way other than multiplying the two sums together. I hope you guys can give me a small push on this. Thanks very much in advance, And have a good day. Getting new files to inherit group permissions on Linux I am having a problem with permissions on a Linux server. I am used to BSD. When a directory is owned by a group the user who owns it isn't in such as www-data, files created in it will be owned by that group. This is important because I want files to be readable by the webserver (which I will not run as root) but so a user can still put new files in the directory. I can't put the users in www-data because then they can read every other users websites.   I want the webserver to read all websites, I want users to be able to change their own.  The permissions are set like this on the folders at the moment....  drwxr-x--- 3 john www-data 4096 Feb 17 21:27 john   It is standard behavior on BSD for permissions to work this way. How do I get Linux to do this? Can a multiclass sorcerer/cleric use the Metamagic class ability with non-sorcerer spells? For example, using Twinned Spell when casting cure wounds to heal two creatures instead of just one? New "[tag:" syntax interferes with Markdown links Links in [link text][number] format don't work properly if they follow a tag using the new [tag:tagName] format. Here's the result of [tag:python] [gold badge page][1]:      [gold badge page]   Putting the tag inside the link is worse. Here's the result of [[meta-tag:bug] some text][2], where [2] is google.com:        The part of the link outside the tag just disappears silently.  EDIT: To be more precise, the entire link to [2] disappears. The markup that is produced contains (invalid) nested anchor tags, as shown here from Firefox's "View Source":  &lt;a href="http://google.com" rel="nofollow"&gt;&lt;a href="/questions/tagged/bug" class="post-tag required-tag" title="show questions tagged 'bug'" rel="tag"&gt;bug&lt;/a&gt; some text&lt;/a&gt;   So the tag is still a link, but it is a link to the tag page, which is the behavior it would have if the link to [2] didn't exist at all. The user's desired behavior in this case is... ambiguous, to be charitable.  EDIT 2: I briefly included markup from Firebug here, but it turned out to be irrelevant. It has now been removed to improve readability. See  if you're interested. Can I open Microsoft Word and Excel files? Can I open Microsoft Word and Excel files with Ubuntu? I am revising for an exam and have just realised that the euclidean algorithm questions in past exams are much harder than in homeworks! So i need some help please.  I have a question here, i already have the solution to it but i dont understand it that well, so i would like someone to please explain what is going on ect, and dont worry about avoiding giving me the full solution as i already have it so a full solution would be good!  Here is the question:  Use euclids Algorithm to find $gcd(172,20)$.  Hence solve $172a + 20b = 1000$ giving all solutions in terms of parameter t. List any solutions for which $a,b&gt;0$.  So it wants the answers in terms of parameter t, which is something i am not familiar with and cannot find in my books.   I have the solution in the solutions paper as   $$172= 8\times20+12 $$ $$20=1\times 12 + 8$$ $$12=1 \times 8 +4$$ $$8=2\times 4+0$$  thus $gcd(172,20)=4$  then   $$4= 12-8=12-(20-12)=2\times 12 - 20$$  $$=2\times(172-8\times20)-20$$  $$=2 \times 172 -17 \times20$$   multiplying by 250 we get   $$1000=500 \times 172 -4250 \times 20$$  so one solution is $a=500, b=-4250$  So, I understand everything up until here!!  Now it says,  Generally, $$a=500-t \frac{20}{4}=500-t$$            $$b=-4250+t \frac{172}{4}=-4250+43t$$  To have $a,b&gt;0$, we need $$500-5t&gt;0, (100&gt;t)$$ to have $a&gt;b$ we need $$-4250+43t&gt;0,(t&gt;98.8)$$$  Need $t=99$, $a=5$, $b=7$  Could someone please help explain what is going on in this last bit of working out? I am pretty good at euclidean algorithm usually but havent seen this type before an am panicking  bit as so close to exam! Any help appreciated how to find t and the other positive solutions would be great!  Please could you show me exactly how it is done here as my lecturer is quite strict with us using his exact methods. Many thanks Zero probability and impossibility I read a comment under :     There are plenty of events that can occur that have zero probability.   This reminds me that I have seen similar saying before elsewhere, and have never been able to make sense out of it. So I was wondering    if zero probability and impossibility mean the same? if an event with zero probability doesn't mean that the event is impossible to occur, how probability theory represents/describes impossibility?   Thanks and regards! I've already read all the pages in this site trying to find the answer to my problem but no one seems to be the right one form me...  First I explain you the kind of data I'm working with...  Let's say that I have an array vector with several names of city, one for each of 300 users. I also have another array vector with scores response to a survey of each user or a continuous value for each user.  I would like to know if exist a correlation coefficient that compute the correlation between these two variables so, between a nominal and a numeric/continuous or ordinal variables.  I've searched on the Internet and in some pages they suggest to use the contingency coefficient or Cramer's V or Lambda coefficient or Eta . For each of this measure the just say that they could be applied for such data in which we have a nominal variable and interval or numerical variable. The thing is that searching and searching, trying to understand every one of them, sometime is written or watching the examples that they are reasonable to use them if you have dichotomous nominal variable, except for Cramer's V, other time is not written any requirement for the type of data. A lot of other pages say that is right to apply regression instead, that is right, but I would just simply like to know if there is a coefficient like pearson/spearman for this kind of data.  I also think that is no so properly to use Spearman Correlation coeff since the cities are not sortable.  I have also built the function of Cramer'sV and Eta by myself (I'm working with Matlab) but for Eta they don't talk about any p-value to see if the coefficient is statistically significant...  In the matlabWorks site there is also a nice toolbox that says to compute eta^2 but the kind of input it needs is not understandable.  Is here someone that have done a test like mine? If you need more detail to understand the kind of data I'm using just ask me and I'll try to explain you better. I had just arrived at the hotel, checked in and gotten into my room. Just as I was about to unpack, I noticed that one of my bags was missing. At first I thought I had left it back at the airport, but it turned out that I had left it on the taxi.  Here, what's the meaning and role of "just"?  Why do you guys input the word "just" here?  What exactly does the word do grammatically?   Could you help me? For some reasons, I would like to do an explicit quoting of a string value (becoming a part of constructed SQL query) instead of waiting for implicit quotation performed by cursor.execute method on contents of its second parameter.  By "implicit quotation" I mean:  value = "Unsafe string" query = "SELECT * FROM some_table WHERE some_char_field = %s;" cursor.execute( query, (value,) ) # value will be correctly quoted   I would prefer something like that:  value = "Unsafe string" query = "SELECT * FROM some_table WHERE some_char_field = %s;" % \     READY_TO_USE_QUOTING_FUNCTION(value) cursor.execute( query ) # value will be correctly quoted, too   Is such low level READY_TO_USE_QUOTING_FUNCTION expected by Python DB API specification (I couldn't find such functionality in  document). If not, maybe Psycopg2 provides such function? If not, maybe Django provides such function? I would prefer not to write such function myself... How can I fix "unbalanced brackets" in my data tag? Often, when making a command block contraption, you need to set an entity's data tag. These tags can sometimes be very long, with many layers of nested curly and square brackets. For example, this is the command to summon a villager with a single custom trade:  /summon Villager ~ ~ ~ {Profession:3,Career:2,Offers:{Recipes:[{buy:{id:minecraft:diamond,Count:6b}},sell:{id:minecraft:diamond_hoe,tag:{ench:[{id:16s,lvl:10s}]}]}}   The single-line command block interface of Minecraft makes it incredibly difficult to find and fix errors, especially when it comes to mismatched brackets. In fact, for illustration purposes, I put an error in the above command.  What techniques can I use to find and fix unbalanced square or curly brackets when writing long and complicated data tags?