Grub and Windows on L
Windows 10 update broke my MBR / GRUB
Evaluate   $$\lim_{n\rightarrow \infty}\frac{1}{2n}\cdot \ln \binom{2n}{n}.$$   $\underline{\bf{My\;\;Try}}::$ Let $\displaystyle y = \lim_{n\rightarrow \infty}\frac{1}{2n}\cdot \ln \binom{2n}{n} = \lim_{n\rightarrow \infty}\frac{1}{2n}\cdot \ln \left(\frac{(n+1)\cdot (n+2)\cdot (n+3)\cdots (n+n)}{(1)\cdot (2)\cdot (3)\cdots (n)}\right)$  $\displaystyle y = \lim_{n\rightarrow \infty}\frac{1}{2n}\cdot \left\{\ln \left(\frac{n+1}{1}\right)+\ln \left(\frac{n+2}{2}\right)+\ln \left(\frac{n+3}{3}\right)+\cdots+\ln \left(\frac{n+n}{n}\right)\right\}$  $\displaystyle y = \lim_{n\rightarrow \infty}\frac{1}{2n}\cdot \sum_{r=1}^{n}\ln \left(\frac{n+r}{r}\right) = \lim_{n\rightarrow \infty}\frac{1}{2n}\cdot \sum_{r=1}^{n}\ln \left(\frac{1+\frac{r}{n}}{\frac{r}{n}}\right)$  Now Using Reinman Sum  $\displaystyle y = \frac{1}{2}\int_{0}^{1}\ln \left(\frac{x+1}{x}\right)dx = \frac{1}{2}\int_{0}^{1}\ln (x+1)dx-\frac{1}{2}\int_{0}^{1}\ln (x)dx =  \ln (2)$  My Question is , Is there is any method other then that like Striling Approximation OR Stolz–Cesàro theorem OR Ratio Test  If yes then please explain here  Thanks I'm trying to get the subcaptions and captions in sanserif font and Uppercased. According the manual of the caption package this should be possible in the following way.However, this doesnt seem to work. Am I defining something in the wrong way here?  \documentclass{article} \usepackage{floatrow} \usepackage[demo]{graphicx} \usepackage{subfig} \usepackage{caption} \floatsetup[figure]{subcapbesideposition = top}  \captionsetup{labelfont={sf,up}, labelsep=period, format = plain, singlelinecheck = false, textfont=rm, skip = 2pt} \captionsetup[subfloat]{labelfont = {up,sf},format = plain,textfont = rm,labelformat = simple, labelsep = period}  \begin{document}   \begin{figure}[h!]   \ffigbox[\textwidth]     {\begin{subfloatrow}         \sidesubfloat[]{\includegraphics[width = 0.4\textwidth]{A.pdf}}%         \sidesubfloat[]{\includegraphics[width = 0.4\textwidth]{B.pdf}}%       \end{subfloatrow}       \begin{subfloatrow}         \sidesubfloat[]{\includegraphics[width = 0.4\textwidth]{C.pdf}}%         \sidesubfloat[]{\includegraphics[width = 0.4\textwidth]{D.pdf}}%       \end{subfloatrow}}     {\caption{this is a caption}} \end{figure}  \end{document} Declaring static data members of normal class and class template Who knows about the summation of this series: $$\sum\limits_{i=1}^{n}i^m $$  where $m$ is constant and $m\in \mathbb{N}$?  thanks How SELECT column does not list in GROUP BY clause The query:  select year,sum(mark),sum(maxmark)  from table1  group by parameterno   yields an error message:     Column 'table1.year' is invalid in the select list because it is not contained in either an aggregate function or the GROUP BY clause. Lost space after removing bootcamp Log out button illogical placement In the previous version the logout button is under your username, which it can be found by clicking the down arrow. However, in the new version, after a couple minutes of searching, it is under the "site-switcher" button:        This is really illogical since the logout button is usually under your username, including Google and Yahoo! services.  I suggest bringing back the  which will definitely benefit new users and even old users. Expected Value of a Hypergeometric Random Variable Why are ammeters used in series and voltmeters in parallel? Proving a relation between inradius ,circumradius and exradii in a triangle Prove that in a triangle $$r^2+r_1^2+r_2^2+r_3^2=16R^2-(a^2+b^2+c^2)$$ where the symbols have their usual meanings. I am looking for a smaller or elegant proof using trigonometry. A geometric proof would be super cool. I did prove it but it took me ages to do the calculation. Thanks. Are split infinitives grammatically incorrect, or are they valid constructs? What does ':' (colon) do in JavaScript? Cauchy-Product of non-absolutely convergent series While grading some basic coursework on analysis, I read an argument, that a Cauchy product of two series that converge but not absolutely can never converge i.e. if $\sum a_n$, $\sum b_n$ converge but not absolutely, the series $\sum c_n$ with $$c_n= \sum_{k=0}^n a_{n-k}b_k$$ diverges.  Although we didn't have any theorem in the course stating something like this, it made me wonder if it was true. Find the maximum value of $ f(x) = \int^{x}_{0} \sqrt{(x^2-x)^2+y^4}~dy$ "Now" vs "by now" Consider these examples:   By now the desk should be looking significantly tidier!  By now, the failure of regulators to contain dangerous forces should be well accepted.  By now I own some forty copies of the magazine.  By now the blue of twilight is glowing.  He used leverage, but by now applied concerted discipline to contain his risks.    I wonder if I can use &quot;now&quot; instead of &quot;by now&quot; to convey the same meaning. I would think &quot;by now&quot; refers to a moment until now, but why not use past tense or present perfect tense in the these examples? Express $\sqrt{3}\sin\theta - \cos\theta$ as: $a\cos (\theta + \alpha) $ Express $\sqrt{3}\sin\theta - \cos\theta$ as: $a\cos (\theta + \alpha) $  Can someone please explain to me how to go about doing this? How to Silence the read command The read command is useful when reading data from the user . I am trying to read a number from 1-5 from the user. How can I make it so the input of the user does not show up in the log More than 1000 gallons of paint is/are sold each day Possible Duplicate:             Can anyone help me determine the correct verb in this sentence? I am not sure what to do. If it were not such a complex introductory phrase, it would be more obvious. The general consensus of my friends who are not professional writers is that the verb should be are.   To me the question is whether or not the subject is singular (i.e., “a large collective volume of paint”, perhaps in a tank) or plural (a lot of the individual gallon containers of paint).  If simplified to other options, it would be like these:   Paint is sold. Gallons of paint are sold. More paint is sold. More gallons of paint are sold. More than 1000 gallons of paint is sold. [emphasis on total volume] More than 1000 gallons of paint are sold. [emphasis on individual containers] ESM is available for 12.04, which enables Advantage customers to keep receiving kernel updates and updates for many user space packages. In our case it's very important to know if we'll be able tot use Ubuntu 14.04 safely after April '19, because it is not possible to upgrade our application and migrate it to say, 16.04. Count below a comment is misleading