Prove that $(1+\frac{1}{n})^k < 1+ \frac{k}{n}+\frac{k^2}{n^2}$
prove that if $1 \le m \le n$ then $\left(1+\frac1n\right)^m < 1+\frac mn + \left(\frac mn\right)^2$
When should I start an LLC for my side work? I am doing quite a bit of sidework this year and wondering at what point would it be a good idea to "start a business" and bill through that. Determining the number of bit strings of length n that contains no adjacent 0s • $C_n =$ this number of bit strings  • A binary string with no adjacent 0s is constructed by   Adding “1” to any string w of length $n-1$ satisfying the condition, or Adding “10” to any string v of length $n-2$ satisfying the condition   I still don't find that intuitive at all. Why adding up $C_{n-1} + C_{n-2}$ results in $C_n$? I have tried my best and asked my friends for help, but they find the formula unintuitive as well. Could anyone here come up with a clear explanation? Thanks! Would time freeze if you could travel at the speed of light? How can I get debug log in Mavensmate I am new to TeX/LaTeX. How do I install TeX on Windows (7)? It seems that there are many different implementations, so which one is best for my OS? It turns out that removing a virtual package with apt-get only remove the package itself,  leaving behind the real packages that were installed by it  So how do you remove a virtual package, along with all the real packages that it installs? What is the difference between DPI (dots per inch) and PPI (pixels per inch)? There are lots of other questions on Graphic Design which partly cover this, e.g. .  However, I have become frustrated at the number of questions and answers which confuse the two terms. I think understanding the difference is important.  So here's a place to answer this question well and clear up the confusion! How to find out why "context is incorrect"? I have been trying to write my first addon for Blender over the last couple of days. One error I came across quite often is the context is incorrect error for an operation of bpy.ops.  I learned that one can ask the poll function of an operator if the context is correct, but this does not tell why the context is correct/incorrect. What would be the best way to figure that out? Only way I could think of is to look into the code of the poll function. If that is the best way, what is the fastest way to find the appropriate piece of code? Can't join a community I look for similar questions to this one but my problem is different. I want to join this community:    so I click in     but I got this error:    but then, I found that the community is already in list of communities:    However, if I click on the community, I'm an anonymous member     and even I have the option to join again     but if I click on it, the problem starts again :( what can I do? Proving the limit of a function of a sequence is equal to the function of the limit of that sequence Suppose $f$ is a continuous function at $x = c$ in $[a,b]$.  Prove that for any sequence ${x_n}$ in $[a,b]$ converting to $c$, the sequence $\{f(x_n)\}$ converges to $f(c)$.  That is, $$ \lim_{n\to\infty}f(x_n)= f\left(\lim_{n\to\infty}x_n\right)$$    This proof seems simple but there are a few things that I need to know first. If $\{x_n\}$ converges to $c$, is it sufficient to substitute $c$ in for $\lim_{n\to\infty}x_n$?  Also needing some guidance on the structure of this proof.  Thanks! Is there a way to connect my MacBook Pro to two external monitors (either VGA or HDMI; no Thunderbolt) in such a way that the Mac screen serves as a third monitor?  What hardware do I need? Which Python memory profiler is recommended? I want to know the memory usage of my Python application and specifically want to know what code blocks/portions or objects are consuming most memory. Google search shows a commercial one is  (Windows only).  And open source ones are  and .  I haven't tried anyone, so I wanted to know which one is the best considering:   Gives most details. I have to do least or no changes to my code. Definition of textbook RSA What is the definition of textbook or "raw" RSA?  What are some of the properties of textbook RSA?  How does it differ from other schemes based on RSA? Search which man sections contain a given entry Selling a car with a lien I own a car personally, but it is mortgaged to a bank. I want to sell it to an individual. I haven't the funds to repay the loan until I'm paid for the car. The buyer doesn't want to pay me without a lien release (or guarantee that one is coming). The bank won't release the lien until it gets the funds. What do people do in a case like this? It must be a common case. What should I do? Prove that $\int_0^\infty \frac{x\,\log x}{(1+x^2)^2} = 0$ I have been asked to prove that $$\int_0^\infty \frac{x\,\log x}{(1+x^2)^2}\,dx = 0$$  I tried proceeding as follows: \begin{align} \int_0^\infty \frac{x\,\log x}{(1+x^2)^2}\,dx &amp;= \int_0^1 \frac{x\,\log x}{(1+x^2)^2}\,dx + \int_1^\infty \frac{x\,\log x}{(1+x^2)^2}\,dx \\ &amp; = \int_0^1 \frac{x\,\log x}{(1+x^2)^2}\,dx + \int_{u=1}^0 \frac{u\,\log u}{(1+u^2)^2}\,du\quad (\text{where }u=1/x) \\ &amp;= \int_0^1 \frac{x\,\log x}{(1+x^2)^2}\,dx - \int_0^1 \frac{u\,\log u}{(1+u^2)^2}\,du \\ &amp;=^? \int_0^1 0 \,dx \\ &amp;= 0 \end{align}  But I think that I should first show that the improper integral $\int_0^1 {x\,\log x}/{(1+x^2)^2}\,dx$ is convergent.  Do I need to show it? If yes, how to do so?  (I have been taught basic methods such as comparison, limit comparison and Dirichlet test.) Loopback to forwarded Public IP address from local network - Hairpin NAT Resizing Mac OS partition - macOS High Sierra Is there a way to make IE8 honour opacity on an `:before` pseudo element? Fancy Chapter Headings Style I would like make a fancy chapter like the figure.    However, i made this code  \documentclass[12pt,a4paper]{book}   \usepackage{lmodern} \usepackage{titlesec} \usepackage{microtype} \usepackage{tikz}    \titleformat{\chapter}[display]   {\normalfont\bfseries\color{black}}   {%\filleft%     \begin{tikzpicture}     \node[       outer sep=-10pt,       text width=2.5cm,       minimum height=3cm,       fill=black,       font=\color{white}\fontsize{80}{90}\selectfont,       align=center       ] (num) {\thechapter};     \node[       rotate=0,       anchor=south,       font=\color{black}\Large\normalfont       ] at ([yshift=43pt,xshift=30]num.west)  {\textls[180]{\textsc{\chaptertitlename}}};       \end{tikzpicture}%  }    {-90pt}   {\titlerule[2.5pt]\vskip3pt\titlerule\vskip4pt\LARGE\sffamily \raggedright }    \begin{document} \chapter{The Singular Sturm-Liouville Problems in Spectral Parameter}    \end{document}      that produces this one    How can I produce the chapter like the First Figure?