evaluate $\lim_{n\to \infty} \left \{\frac{1}{k.n}+\frac{1}{k(n+1)}+\frac{1}{k(n+2)}+...+\frac{1}{k(n+p-k)} \right \}$    i think it can be written as a integral somehow but i can't
There are four options. Which of them is correct?- (a)${p\over k}$ (b)${k\over p}$ (c)$\log({p\over k})$ (d)${\log p\over k}$ I somehow want to use the rule for evaluating this kind of sum using integration, i.e. $\int_{0}^{1} f(x) \ dx=\lim_{n\to\infty} {1\over n}\sum_{r=1}^{n}f({r\over n})$  But the given expression $\lim_{n\to\infty} [{1\over kn}+{1\over k(n+1)}+{1\over k(n+2)}+\cdots+{1\over k(n+p-k)}]={1\over k}\lim_{n\to\infty}{1\over n}\sum_{r=0}^{p-k}\frac{1}{1+{r\over n}}$, sum running from $0$ to $p-k$ instead of $n$. So, how to evaluate this limit? Thanks for assistance in advance.
what does ∇ (upside down triangle) symbol mean in this problem Given $f(x) = \frac{1}{2}x^TAx + b^Tx + \alpha $   where A is an nxn symmetric matrix, b is an n-dimensional vector, and alpha a scalar. Show that  $\bigtriangledown _{x}f(x) = Ax + b$   and  $H = \bigtriangledown ^{2}_{x}f(x) = A$   Is this simply a matter of taking a derivative with respect to X, how would you attack this one? proof of binomial identity involving double sum Why java classes do not inherit annotations from implemented interfaces? I have seen several answers and proofs here at MSE that a self-adjoint operator $T$ on a Hilbert space $H$ where $T^2=T$, implies $T$ being an orthogonal projection operator.   The thought struck me; what would happen if the condition $T^2=T$ is replaced by $T^3=T^2$ or instead by $T^4=T^3$ or why not more generally by $T^n=T^{n+1}$ for some $n\in\mathbb{N}$? Would this still imply $T$ to be an orthogonal projection operator?  I guess someone must have thought of this before me? In that case perhaps there is a reference in the literature for this or perhaps the proof is way simpler than I might expect? Could $X^3 + X + 1$ have a root in $E?$ Consider the irreducible polynomial $g = X^4 + X + 1$ over $F_2$ and let $E$ be the extension of $F_2 =$ {0, 1} with a root $α$ of $g.$  Could $X^3 + X + 1$ have a root in $E?$  When working through this sheet Ive come across this question in which im not sure how to answer so any help will be appreciated If a series converges, then the sequence of terms converges to $0$. Why do universal constants have the values they do? Is it acceptable to say "fed up OF" - surely it should be "fed up WITH" or "bored WITH something" rather than "bored OF something"? Unable to mount ISO in write mode How can I get TeXstudio's syntax highlighting to recognize an unknown command? TeXstudio does not recognize the LaTeX command \rm (red highlighting in the document, with the message: unrecognized command), even after I activate all possible completion files (.cwl).  In such a case, what can I do so that TeXstudio recognizes a certain command for highlighting? How to throttle callback of jQuery event? Explode string into array with no empty elements? PHP's explode function returns an array of strings split on some provided substring. It will return empty strings when there are leading, trailing, or consecutive delimiters, like this: var_dump(explode('/', '1/2//3/')); array(5) {   [0]=&gt;   string(1) &quot;1&quot;   [1]=&gt;   string(1) &quot;2&quot;   [2]=&gt;   string(0) &quot;&quot;   [3]=&gt;   string(1) &quot;3&quot;   [4]=&gt;   string(0) &quot;&quot; }  Is there some different function or option or anything that would return everything except the empty strings? var_dump(different_explode('/', '1/2//3/')); array(3) {   [0]=&gt;   string(1) &quot;1&quot;   [1]=&gt;   string(1) &quot;2&quot;   [2]=&gt;   string(1) &quot;3&quot; } I've just purchased a new brand laptop with 17 inch size of the screen. I've installed Texmaker and it is messy. The icons are very small and the GUI is not zoomed out automatically to fit the contents of the window. These are pictures         I've already set the size of the icons to medium and the size of apps to 250 (i.e. the recommended size) for  the display.  I'm using Windows 10. Some fonts are blurry and a lot of people are complaining about this problem with no solution from Microsoft. Any suggestions? Why can't I connect to Pokemon Go via mobile data? Android app chokes on per-site meta profiles Steps to reproduce:    Go to a per-site meta. Find any post. Touch the user that posted it. See something like this (click to enlarge):           As you can see, it says "Stack Overflow" as the site, and shows my SO rep and badges, but it shows my Meta.Code Golf posts. How can I split multiple joined words? I have an array of 1000 or so entries, with examples below:  wickedweather liquidweather driveourtrucks gocompact slimprojector   I would like to be able to split these into their respective words, as:  wicked weather liquid weather drive our trucks go compact slim projector   I was hoping a regular expression my do the trick.  But, since there is no boundary to stop on, nor is there any sort of capitalization that I could possibly key on, I am thinking, that some sort of reference to a dictionary might be necessary?    I suppose it could be done by hand, but why - when it can be done with code! =)  But this has stumped me.  Any ideas? Much effort has been expended on a famous unsolved problem about the .  Not surprisingly, it's called the Riemann hypothesis, which asserts:  $$ \zeta(s) = 0 \Rightarrow \operatorname{Re} s = \frac1 2 \text{ or } \operatorname{Im} s = 0 .$$  Now there are numerical methods for approximating $\zeta(s)$, but as I understand, no one knows any exact values except at even integers, which include the trivial zeroes for which $\operatorname{Im} s = 0 $.  So I've always wondered:  For the rest, how does one prove $\operatorname{Re} s = 1/2$ exactly?  (All I know that seems vaguely useful is the argument principle, which I'm not sure helps here, but I'd be happy to learn about other techniques that aren't too far advanced.)  Edit:  Looking over this again, I found out I missed the closed-form values at .  This doesn't affect the question but seemed worth correcting.  Thanks to the people who contributed. A question about a primitive root mod $p=2^{2^k}+1$, where $p$ is prime. $100$-th derivative of the function $f(x)=e^{x}\cos(x)$ I've got this task I'm not able to solve. So i need to find the 100-th derivative of $$f(x)=e^{x}\cos(x)$$ where $x=\pi$.  I've tried using Leibniz's formula but it got me nowhere, induction doesn't seem to help either, so if you could just give me a hint, I'd be very grateful.  Many thanks! Energy conservation in General Relativity I understand that energy conservation is not a rule in general relativity, but I'd like to know under what circumstances it can still be possible. In other words, when is it possible to associate a potential energy to the gravitational field, so that the energy is constant in the evolution of the system?  Here are some examples, is there a convenient way to define energy in these scenarios?   Just a system of gravitational waves. A point mass moving in a static (but otherwise arbitrary) space-time. Equivalent (if I'm not mistaken) to a test mass moving in the field of a second much larger mass, the larger mass wouldn't move. Two rotating bodies of similar mass.   Overall, I'm trying to understand what keeps us from associating a potential energy to the metric. When we break the time translation symmetry of a system by introducing an electromagnetic field, we can still conserve energy by defining an electromagnetic potential energy. Why can't we do the same when we break TT symmetry by making space-time curved?