I know I can compare  $\sum_{n \leq x}1/n$ with $log(x).$ I also think $log(x) \leq\sum_{n \leq x}1/n.$
Consider the partial sum of the divergent Harmonic series $H_n = \sum\limits_{k = 1}^{n}\frac{1}{k}$. I recently saw a question which required finding out the asymptotic bounds of $H_n$. Now, I could not find any bound closer than $O(n)$. So I looked up the series and found  Wikipedia article which says that $H_n - \ln n$ approaches a constant (the ). This clearly shows that $H_n = \Theta(\log n)$.  However, considering the question was in an beginners undergraduate algorithms course in CS, I was wondering if there is a simpler proof of this? Is there some clever proof requiring elementary mathematics to show $H_n = \Theta(\log n)$?  PS: This is not homework in the sense that it is not my homework, and the actual question does not ask for a proof, but rather asks a lot of functions to be arranged asymptotically.
dd writing on USB flash drive causes massive reads Is there a way to achieve the equivalent of a  in javascript regular expressions?  I need to match a string that does not start with a specific set of characters.  It seems I am unable to find a regex that does this without failing if the matched part is found at the beginning of the string.  Negative lookbehinds seem to be the only answer, but javascript doesn't have one.  EDIT: This is the regex that I would like to work, but it doesn't:  (?&lt;!([abcdefg]))m  So it would match the 'm' in 'jim' or 'm', but not 'jam' I've got this one file, favicon.ico. It's actually a large group of nested files, but purely for example, just this one.  If I do cp favicon.ico favicon2.ico, Apache can read the second one just fine.  If I do mv favicon.ico favicon3.ico, Apache throws a 403 on favicon3.ico.  If I su apache, I can read the file as apache.  stat says both files are identical, except for the inode number.  Is there some hidden flag somewhere or something that I'm missing? I've never seen anything like this.  error_log says (13)Permission denied: access to /favicon3.ico denied Selling customized versions of Ubuntu My question is next: "Can I modify some version of Ubuntu Linux, add some software related to electrical engineering, modify look and some functions of Ubuntu, basically customize the entire system so it suits the needs of electrical engineers working on the field or in the office, and then that customized system sell in order to make profit?"   I'm not sure am I allowed to do that (is it legal?), so I would appreciate answer. A function such that $f:\mathbb C \to \mathbb C$ such that $f(z)f(iz)=z^2$ satisfies $f(z)+f(-z)=0$.(MADHAVA-2020) Suppose $f$ is a function such that $f:\mathbb C \to \mathbb C$ such that $f(z)f(iz)=z^2$, then we have to show that $\forall z\in \mathbb C f(z)+f(-z)=0 $.   Actually I have solved this problem and have no doubt about it but actually this question came in MADHAVA-2020. So I posted it on this site.   I am not answering the question and inviting all users to answer it.  It is a nice problem.   I will answer it within a week, but I am letting everyone try. Proving a function is constant, under certain conditions? The problem:  Assume $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfies $|f(t) - f(x)| \leq |t - x|^2$ for all $t, x$. Prove $f$ is constant.  I believe I have some intuition about why this is the case; i.e. if $t$ and $x$ are very close, ($|t - x| = \epsilon$), then $\sqrt{\epsilon}$ will converge to 0, so $|f(t) - f(x)|$ will also converge to zero if you keep taking $t$ and $x$ closer and closer to each other. However, how do I formalize this argument? Thanks. Show that $8 \mid (a^2-b^2)$ for $a$ and $b$ both odd I know there is something wrong with this but I don't know where. It's some kind of a math fallacy and it is driving me crazy. Here it is: $$-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1?$$ How extended must my extended paper be? I would like to set up memoir so that when I use \footnote{foo}, foo is displayed in the margin with the proper label and preferably break over pages.  There is an option \footnotesinmargin which gives almost exactly what I need, only justified. I want it to use ragged2e and change alignment based on even/odd page so that the margin note is always flushed towards the main text area and allows hyphenation on the side of the edge.  \documentclass[12pt, draft]{memoir} \footnotesinmargin \usepackage{polyglossia} \setmainlanguage{english} \usepackage{fontspec} \defaultfontfeatures{Ligatures=TeX} \setmainfont[Ligatures=TeX,Numbers=OldStyle]{Minion Pro} \usepackage{kantlipsum}  \begin{document} \kant[1]\footnote{\kant[1]} \kant[2]\footnote{\kant[2]} \kant[3-5] \end{document}   Possible partial work-around  I have remembered a nice thesis based upon the Classic thesis template and found the  which indeed contains a possible solution:  \usepackage{marginnote} \usepackage{ragged2e} \newcommand{\sidemark}[1]{#1.{\:}}  \newcommand{\marginelement}[2][0]{%   \marginnote{%     \strictpagechecktrue     \checkoddpage     \ifoddpage%       \RaggedRight\footnotesize%     \else%       \RaggedLeft\footnotesize%     \fi%     #2%   }[#1\onelineskip]% }%  \newcommand{\sidenote}[2][0]{%   \footnotemark%   \ignorespaces%   \marginelement[#1]{%     \sidemark{\thefootnote}%     \ignorespaces#2%   }%   \unskip% }%   Nevertheless, I would still prefer a memoir-specific solution with the re-definition of footnotes when used with the option \footnotesinmargin mainly for the sake of portability. This work-around also does not work with hyperref (there is no hypertext connection) and does not break over pages automatically. The following MWE produces a PDF whose TOC has two entries: “Foo” and “Bar” with correct page numbers. So far, so good. However, upon clicking on the link to “Bar”, I am taken to the page for “Foo” instead.  \documentclass{report} \usepackage{hyperref} \newcommand*\backmatter{\setcounter{chapter}{0}}  \begin{document} \tableofcontents \chapter{Foo} \backmatter \chapter{Bar} \end{document}   (Produced with pdflatex, run twice.)  This MWE is whittled down from a (much!) more complex document but the above does seem to reproduce the error reliably.  Re-setting the chapter count seems to confuse hyperref, even though the page references are still accurate. Notice that in my real document, \backmatter does other things, including setting the numbering style to alphabetical so there’s no chance of confusion even though both chapters are ostensibly labelled as “Chapter 1”.  How do I fix this? Is this a bug? Are there workarounds? How to pass parameters to query? On company 401k matching. Clarification needed I realized I don't fully understand something. It almost embarrassing to admit.  In 2016 we have 18k max contribution limit for 401k. Does this mean that employee can contribute at most 18k or that total contributions made on behalf of employee should not exceed 18k (company match)  To make is simple, if you make 100k, by contributing 18% you got your limit. But if your company matches 5%, actual contributions will be 18.9k. In the above example, if one wants to maximize benefit should one contribute 18% or less? What is the best way to get security token of a API Only user? API only users cannot login and choose 'Reset my security token'. Right now, in my company, we select a System Administrator profile for users, log in for the first time, choose 'Reset my security token', then change the profile to a custom profile which has API only enabled.  The other way of doing it is, modifying the profile to not have API only user checked initially and then turn it on once the security token is received.  Is there a recommend way to do this? Is there an option to force the security token to be sent  when the user is created with a profile which has API user enabled? Matrices - Conditions for $AB+BA=0$ The Problem Let $A$ be the matrix $\bigl(\begin{smallmatrix}a&amp;b\\c&amp;d\end{smallmatrix} \bigr)$, where no one of $a,b,c,d$ is $0$. Let $B$ be a $2\times 2$ matrix such that $AB+BA=\bigl(\begin{smallmatrix} 0&amp;0\\ 0&amp;0 \end{smallmatrix} \bigr)$. Show that either   $a+d=0$, in which case the general solution for $B$ depends on 2 parameters, or $ad-bc=0$, in which case the general solution for $B$ depends on one parameter.   (this is question 22 of the last matrix exercise of Further Pure Mathematics by Bostock et al.)  Comments Writing $B=\bigl(\begin{smallmatrix} e&amp;f\\ g&amp;h \end{smallmatrix} \bigr)$ and multiplying out I get that    $(a+d)(f+g)+(b+c)(e+h)=0$ $ae+bg+cf+dh=0$   but I am unable to get the required restrictions on $a,b,c,d$. Is there a quick way of doing the problem that doesn't require manual computation? I thought of considering invertible and non-invertible cases but couldn't get anywhere. Help would be much appreciated. Wireframe iOS design apps for iPad? Is there an app for iPad to assist in wireframe design for iPad and iPhone apps?  I don't mean actual development, just placing standard widgets and the things you usually do with photoshop and associated kits. Induction on a sum The left hand side has terms involving  $\binom{n}{m}= \dfrac{n!}{(n-m)!m!}$    $$1+\dfrac{1}{2}\binom{n}{1} +\frac{1}{3}\binom{n}{2}+...........+\frac{1}{n+1}\binom{n}{n} = \dfrac{2^{n+1}-1}{n+1}$$  I've done induction and proved $P(0)$ holds and also $P(1),P(2)$ holds (Just in case)  Now I've found $P(K+1)$ so the sum on the left is going to equal $\dfrac{2^{n+1}-1}{n+1}$ + $\dfrac{1}{n+2}$  and the right hand side is going to equal $\dfrac{2^{n+2}-1}{n+2}$  My problem is coming with the algebra though. I can't get those sides to equal. I can't get rid of the $n+1$ in the denominator.   Anyone have any ideas for me? They'd be much appreciated!  P.S. Someone gave me a hint and said that you can use the binomial theorem to solve this. iOS 12 tapping on notification to open app not working Is there a documentclass that produces 'endless' pages? Does anybody know of a documentclass that produces a page that has a fixed width and a variable length, depending on the actual content that needs to fit on the sheet? I'm trying to get something like a kind of scroll if possible. Limit of $\left(1-\frac{1}{n^2}\right)^n$