What is the summation of this infinite series $$1 + \frac{1}{3} + \frac{1}{7} + \frac{1}{15} + \frac{1}{31} + \frac{1}{63} + ... ?$$
How to evaluate this infinite sum?  $$\sum_{n=1}^{\infty}\frac{1}{2^n-1}$$
Recovering a lost website with no backup? How to get Beamer Math to look like Article Math In a LaTeX article, the math characters are more "curvy" than the text characters:  \documentclass[12pt]{article} \begin{document}  $x^2 + y^2 = z^2$ \end{document}     In a beamer presentation, the math characters do not have the same curves:  \documentclass{beamer} \begin{document} \begin{frame}  $x^2 + y^2 = z^2$ \end{frame} \end{document}     Question  How can I get the beamer math to look just like the article math?  I would prefer if the solution was a single line (that I could place at the top of my LaTeX file) as opposed to something that needs to appear next to every piece of math. It is always told as a fact without explaining the reason. Why do two objects get charged by rubbing? Why one object get negative charge and other get positive ? Does the OP of a flagged post get comment notifications if the post is deleted by mods? For symmetric operators one has: $$A\text{ symmetric}:\quad\mathcal{R}(A\pm\imath)=\mathcal{H}\implies A^*=A$$ How to prove this in an unveiling way? Virtual vs Real image Easiest way to delete a column? I have a set of 10 tables and each has an extra column that I don't want. Specifically, these tables are descriptions of database tables exported by phpMyAdmin, and they all have a "MIME" column that I would like to remove. Is there a way to do this other than using string manipulation / find and replace?   I have tried setting the column to p{0in}, but the column title still remains. What is the best way to check whether a given object is of a given type? How about checking whether the object inherits from a given type?  Let's say I have an object o. How do I check whether it's a str? Why isn't sizeof for a struct equal to the sum of sizeof of each member? Why does the sizeof operator return a size larger for a structure than the total sizes of the structure's members? It seems that there are many ways to prove the Sylow theorems. I'd like to see a collection of them. Please write down or share links to any you know. How to get straight quotation marks? How do I get straight quotation marks, i.e., " and ' in TeX?  (No, \verb changes the font and this is unacceptable.)  [Edit]  I forgot to mention that I’m using XeLaTeX, and fontspec is a necessity.  The first two proposed answers did work, without fontspec. How to translate and rotate the heading of landscaped pages? Prove that ${\left(\frac{x+1}{2}\right)}^{x+1} \leq x^x$ I want to prove that for each real number $x&gt;0$, the statement  $${\left(\frac{x+1 } {2}\right)}^{\!x+1} \leq x^x$$  is correct.   I’ve tried using mathematical induction methods, but haven’t found a solution. Although, using any method is appreciated. Suppose $G$ is a non-abelian group and $H,K$ are two abelian subgroups of $G$. Then must $HK$ be an abelian subgroup of $G$?  I know an example, but I am confused. Thus I just want to check that. Question is :  Suppose $Y\subset X$ and $X,Y$ are connected and $A,B$ form separation for $X-Y$ then, Prove that  $Y\cup A$ and $Y\cup B$ are connected.  What i have tried is :   Suppose $Y\cup A$ has  separation say $C\cup D$ then all i could see is that   Either $Y\subset C$ or $Y\subset D$ as $Y$ is connected and $ C\cup D$ is separation for a subset that contains $Y$.  Without loss of generality we could assume $Y\subset C$  As $Y\cup A=C\cup D$ and $Y\subset C$  i can say $D\subset A$ (I do not know how does this help)  I have $X-Y=A\cup B$ with $A\cap B=\emptyset$  I have not used connectedness of $X$ till now, So i thought of using that and end up with following :   $X-Y=A\cup B\Rightarrow X=Y\cup A\cup B=(Y\cup A)\cup (Y\cup B)$  I do not know what to conclude from this...   I would be thankful if some one can help me by giving some "hints"  Thank you I wish to travel to the US and think I am eligible to do so under the Visa Waiver Program. I found a lot of information out there but am still having some trouble understanding how this works.  What is the US Visa Waiver Program? What is an ESTA? Normal + Connected -> Uncountable Here's a solution (to the exercise "prove that if $X$ is a space with more than one element, and is normal and connected then $X$ is uncountable"):   by Urysohn's lemma, given $A$ and $B$, closed and disjoint in $X$, there exists a continuous function from $X$ into $[0,1]$ such that $f(A)={0}, f(B)={1}$.   If $X$ were countable, so it would be $f(X)\subset [0,1]$; choose $r\in (0,1)\setminus f(X)$, then $X=f^{-1}([0,r))\cup f^{-1}((r,1])$, so $X$ is disconnected.  But the full power of the lemma wasn't used, all that was used is that there exists a continuous function $X\longrightarrow [0,1]$, so the condition of being normal seems too much.  So my question is, what are the weakest conditions on $X$ for the existence of such continuous functions into $[0,1]$ (or $\mathbb{R}$, for that matter)? Why don’t photons interact with the Higgs field and hence remain massless? Last digits, numbers Can anyone please help me?  1) Find the last digit of $7^{12345}$   2) Find the last 2 digits of $3^{3^{2014}}$.  Attempt: 1)  By just setting the powers of $7$ we have $7^1 = 7$, $7^2=49$, $7^3=343$, $7^4 = 2401$, $7^5 = 16807$, $7^6 = 117649$, $\dots$  After the power of $4$, the last digits will repeat. Then by noticing the pattern the digits will end in $7,9,3$ and $1$.  Then we can divide the exponent $(12345)$ by $4$ since this is the cycle that makes it repeat. Then $12345 : 4$ has remainder $1$. So $7^1 = 7$ is the unit digit to $7^{12345}$.   So the last digit is $7$.   I know how to do it like this, the problem does not state how to find the last digit, but I know it has something do do with Euler's theorem.  for part $2$) I don't know how to start. Can anyone please help me?   Thank you for the help. Resolve build errors due to circular dependency amongst classes