Limit of a Riemann sum $\lim_{n\to \infty} \frac{1^p+2^p+\ldots+n^p}{n^{p+1}}$
What is the result of $ \lim_{n \to \infty} \frac{ \sum^n_{i=1} i^k}{n^{k+1}},\ k \in \mathbb{R} $ and why?
What is the exposure triangle? How do the "sides" affect my photographs? How to create Mist in Blender Find $\int \frac{x^3}{x^2-1} \:\mathrm{d}x$ Find $\displaystyle \int \dfrac{x^3}{x^2-1} \:\mathrm{d}x$.   I think there are a few ways to do this. We could just break this up using long division or break it up into partial fractions then use long division. My homework says to break this up into its roots so that's why I brought up partial fractions. Is that the best way to do it here? How can I pass arguments to a batch file? I need to pass an ID and a password to a batch file at the time of running rather than hardcoding them into the file.  Here's what the command line looks like:  test.cmd admin P@55w0rd &gt; test-log.txt Notifications in browser tab A very nice feature for Stack Overflow (and other Stack Exchange sites) would be to indicate the number of notifications (if any) in the browser tab like seen on Facebook:    Of course that would require automatic update if content has changed. Some kind of 'push-notification' which would be a nice feature as well. How to correctly setup Ubuntu 18.04 LTS installed from mini.iso to use NetworkManager? Colorizing your terminal and shell environment? I spend most of my time working in Unix environments and using terminal emulators. I try to use color on the command line, because color makes the output more useful and intuitive.  What options exist to add color to my terminal environment? What tricks do you use? What pitfalls have you encountered?  Unfortunately, support for color varies depending on terminal type, OS, TERM setting, utility, buggy implementations, etc.  Here are some tips from my setup, after a lot of experimentation:   I tend to set TERM=xterm-color, which is supported on most hosts (but not all). I work on a number of different hosts, different OS versions, etc. I use everything from macOS X, Ubuntu Linux, RHEL/CentOS/Scientific Linux and FreeBSD. I'm trying to keep things simple and generic, if possible. I do a bunch of work using GNU screen, which adds another layer of fun. Many OSs set things like dircolors and by default, and I don't want to modify this on a hundred different hosts. So I try to stick with the defaults. Instead, I tweak my terminal's color configuration. Use color for some  (ls, grep, less, vim) and the . These commands seem to use the standard "". For example:  alias less='less --RAW-CONTROL-CHARS' export LS_OPTS='--color=auto' alias ls='ls ${LS_OPTS}'    I'll post my .bashrc and answer my own question Jeopardy Style. How do you make sure email you send programmatically is not automatically marked as spam? This is a tricky one and I've always relied on techniques, such as permission-based emails (i.e. only sending to people you have permission to send to) and not using blatantly spamish terminology.  Of late, some of the emails I send out programmatically have started being shuffled into people's spam folder automatically and I'm wondering what I can do about it.  This is despite the fact that these particular emails are not ones that humans would mark as spam, specifically, they are emails that contain license keys that people have paid good money for, so I don't think they're going to consider them spam  I figure this is a big topic in which I am essentially an ignorant simpleton. Is there any "superlogarithm" or something to solve an equation like this:  $$x^x = 10?$$ In most of the mechanics textbooks , the  is stated without any justification or an experiment to confirm this law. Also after introducing this law forces are also decomposed without any justification. Please suggest any links or books related to this. Any help will be appreciated. Thanks in advance. How can I increase the JVM memory? After having read various articles on this site about WWW or Non-WWW website setup, ,  and , and after having setup a correct redirection rule in .Htaccess to at least eliminate duplicate contents (currently the non-www solution (e.g. http://example.com) I have a feeling that my site rank less since I changed this.   My confusion was strengthened by these two opponents:     &amp;     Could the change of www.example.com >> http://example.com have a factual impact on ranking?  say either is better or worse. Are they holding information back?  I am confused fellow folks...  As much as I would like to separate facts from fiction, I am aware that not all the good valid information is written. Not on google webmaster site and perhaps not elsewhere. If so it seems out of date!  Does or does not a WWW domain impact SEO/rank better than its NON-WWW version? (Given that you have setup a proper redirect to avoid duplicates and go with one) How do "pet" Walkers help ward off other Walkers? In both the comics and the show, Michonne first appears with two walkers that she essentially keeps as pets.    The comics sort of half-explain this - as per :     She notices walkers don't attack one another and hacks off the arms and lower jaws of Mike and Terry. She chains them up by their necks to use them as escorts for an escape that was successful.   In , we see that even if they still sort of smell like Walkers, the Walkers can smell/detect the living and will attack.  In ,      We see Andrea effectively making a new "pet" so that she can use to deter the Walkers that she comes across on her trip to the prison. Along the trip, she is shown to come within close range of Walkers, but they give her space when she points the "pets" in the right direction.   Why would the Walkers not just attempt to go around the "pets"? What about them creates an effective protective barrier? In , someone suggested in a  that I should not cast the result of malloc. i.e., I should do this: int *sieve = malloc(sizeof(int) * length);  rather than: int *sieve = (int *) malloc(sizeof(int) * length);  Why would this be the case? Suppose we have an object in space. say gravity is negligible, and the thing has a rocket on both sides. the thing fires one rocket for, say, 5 seconds. now it's moving. The object has kinetic energy. Now the other rocket fires for 5 seconds. Then it's stationary. where does the energy go? I know that some of the energy from the rockets would just go into space as heat and kinetic energy of the hydrogen particles flying out, but how does there seem to be more energy in the situation if the first rocket just goes for 10 seconds than if one goes for 5 and the other goes for 5? Alignment issue in users page while searching user With $x+y\ge z$ $(x,y,z\ge0)$, prove that: $$\frac{x}{1+x}+\frac{y}{1+y}\ge\frac{z}{1+z}$$  I'm aware that using analytic view this is easy since $f(x)=\frac{x}{1+x}$ is concave in $[0,\infty)$. However I want to prove it using merely algebraic techniques. Is that possible? Showing that if a subset of a complete metric space is closed, it is also complete Let $(X, d(x,y))$ be a complete metric space. Prove that if $A\subseteq X$ is a closed set, then $A$ is also complete.  My attempt: I tried to prove that every Cauchy sequence $(b_n)$ of points of $A$ converges to a point $b\in A$. However could not figure out the exit way. Maybe I am on the wrong track. Could you please help me?  edit: More from my attempt:  Suppose $A$ is a closed set and let $(x_n)$ be a sequence of points $A$ such that $\lim_n x_n\to b$.  Suppose now that $A$ has the property that $b\in A$, whenever $x_n$ converges to $b$. We know that every element of $x_n$ which is convergent in $X$ also converges to a point in $A$. Since $x_n$ is a Cauchy sequence in $A$, it must converge to a point $y\in A$. But the limit of a convergent sequence is unique.   Take $x\in A$ and select an appropriate $n$, which enables $x_n$ to converge to a point $x$ in $X$. Since the limit is unique, it must follow that $x=y$. Thus $x\in A$ and $A$ is closed.   If $A$ is a closed subset of $X$, then any Cauchy sequence of a point in $A$ is convergent in $X$ and hence converges to a point in $A$. Thus $A$ is complete. What's the right way to represent phone numbers? Per User Named Credential and External Identity Users My question is almost similar to one mentioned in 'Auth Provider with per-user Named Credential' (). I am trying to authenticate and make a API call when a new or existing Salesforce community users try to log in. The Authentication is provided by 3rd party (Cognito). So, the requirement is :  User to be authenticated using the 3rd party system for login. After successful authentication and before community page is displayed, the API call is to be made to 3rd party system to get more details about the user.  In order to achieve this:  Created an Open ID Connect Auth. Provider Use the registration handler on the Auth. Provider to create/link a new/existing user. Use a login flow for API call. Use Per User Identity Type Named Credential to make API calls.  No, where I am having issue is once the user has authenticated successfully, how do I allow the user to make API call without authenticating it again. I am getting the error: An Apex error occurred: System.UnexpectedException: You don't have permission to view this data, enter credentials for the named credential in your personal settings. Or ask your Salesforce admin for help. I believe this error is due to the fact that the user is not listed under 'Authentication setting for External Systems'. My requirement doesn't allow me to authenticate the user again, so how do I achieve this using either APEX or any other way?