Show : $\sum\limits_{n=1}^\infty \frac{a_n}{1+a_n}$ is absolutely convergent
Show the series $a_n/(1+a_n)$ converges absolutely
Was Yoda wrong about the limits of telekinesis? In Empire Strikes Back, Yoda  that an object's size doesn't matter and that the difficulty of levitating an X-Wing was only in his mind. If the upper limits of telekinesis are determined by belief, what's to stop a Jedi/Sith from pulling a moon out of orbit, moving an entire solar system, or splitting an atom with the Force? How can I search in another language? Well, this question has been puzzling me for kinda long time, many people believe that orbiting astronauts feel weightless because they are "beyond the pull of Earth's gravity"...How far from the Earth would a spacecraft have to travel to be truly beyond the Earth's gravitational influence? If a spacecraft were really unaffected by Earth's gravity would it remain n orbit? If so, what is the real reason for weightlessness in orbit? Is it possible to hide curves' guides in edit mode? Given that $W$ is a subspace of $V$, prove: $\dim W=\dim V \implies W=V$ Suppose $W$ is a subspace of the finite-dimensional vector space $V$ and $\dim W = \dim V=n$  Let $\{\overrightarrow w_1,\ldots,\overrightarrow w_n\}$ be a basis for $W$ and $\{\overrightarrow v_1,\ldots,\overrightarrow v_n\}$ be a basis for $V$.  The set $\{\overrightarrow w_1,\ldots,\overrightarrow w_n,\overrightarrow v_1,\ldots,\overrightarrow v_n\}$ still spans $W$. Since $W$ is a subspace of $V$, $W\subset V$ and all the $w_i$'s can be written in terms of $\{ \overrightarrow v_{1},\ldots,\overrightarrow v_n\}$.  $\therefore$ We can remove one by one all the $w_i$'s from the set $\{\overrightarrow w_{1},\ldots,\overrightarrow w_n,\overrightarrow v_1,\ldots,\overrightarrow v_n\}$ while still having it span $W$, which leaves us with the set $\{\overrightarrow v_1,\ldots,\overrightarrow v_n\}$. We know it to be linearly independent since it is a basis for $V$. Therefore the basis of $V$ is also a basis of $W$. From there on, it is trivial to prove double inclusion $W\subset V$ and $V\subset W$ which gives us $W=V$.  Is my approach correct? Android SQLite database: slow insertion Do Indian passport holders with UK Tier 2 need to get a tourist visa for Mexico? I'm an Indian passport holder currently resident in the UK under a Tier 2 (General) visa and planning on going to Mexico on a holiday. So I was looking into the rules on the Mexican embassy in the UK and :     According to the new regulations who entered into effect on 18th, May   2016 (article 26 of the Guidelines and Requirements for Migratory   Services) regarding foreign nationals visiting Mexico, holders of a   valid UK visa, regardless their nationality, are not required to apply   for a Mexican visa as a tourist, business (Non-Lucrative Activities)   or transit visitors for a stay of up to 6 months. This regulation does   not apply to British Travel Document holders.   However, :     PERMANENT RESIDENT IN THE UK (ILR / ILE / PERMANENT RESIDENTS)        According to Mexican regulations which came into force in June 2009,   Permanent residents in the United Kingdom wishing to travel to Mexico   do not require a visa to enter the country as tourists or business   visitors for up to 180 days and as visitors in transit for up to 30   days, regardless their nationality.   So which one is correct? Are Tier 2 holders with an Indian passport allowed visa-free / visa on-arrival entry to Mexico? Ubuntu boots from USB but get GRUB rescue after installation I can boot Ubuntu from USB, but when I took the USB stick out of the laptop and booted again, I got an error saying   Error: unknown filesystem. grub rescue&gt;   What can I do to fix this? Proof of the square root inequality $2\sqrt{n+1}-2\sqrt{n}<\frac{1}{\sqrt{n}}<2\sqrt{n}-2\sqrt{n-1}$ How to draw a shaded sphere? Change \textwidth and \textheight in mid-document I set my \textwidth and \textheight in the preamble and would like to change both for some pages. How can this be done? E.g., the following example for \textwidth appears to work for the page number, but not for the text:  \documentclass{article}  \textwidth=5cm \textheight=5cm \begin{document} A \hfill B \eject \textwidth=10cm A \hfill B \end{document}   Output (true to scale): How can I write a program that can detect by itself that it has been changed? I need to write a small program that can detect that it has been changed. Please give me a suggestion!  Thank you. I'm trying to do something like the following:  enum E;  void Foo(E e);  enum E {A, B, C};   which the compiler rejects.  I've had a quick look on Google and the consensus seems to be "you can't do it", but I can't understand why.  Can anyone explain?  Clarification 2: I'm doing this as I have private methods in a class that take said enum, and I do not want the enum's values exposed - so, for example, I do not want anyone to know that E is defined as  enum E {     FUNCTIONALITY_NORMAL, FUNCTIONALITY_RESTRICTED, FUNCTIONALITY_FOR_PROJECT_X }   as project X is not something I want my users to know about.  So, I wanted to forward declare the enum so I could put the private methods in the header file, declare the enum internally in the cpp, and distribute the built library file and header to people.  As for the compiler - it's GCC. When should static_cast, dynamic_cast, const_cast and reinterpret_cast be used? Deciding whether two metrics are topologically equivalent in the space $C^1([0,1])$ Consider the space $C^1([0,1])$ and the function $d:C([0,1])\times C([0,1]) \to \mathbb R$ defined as $d(f,g)=|f(0)-g(0)|+sup_{x \in [0,1]}|f'(x)-g'(x)|$. Decide whether the metrics $d$ and $d_{\infty}$ are topologically equivalent in $C^1([0,1])$ (where $d_{\infty}=sup_{x \in [0,1]}|f(x)-g(x)|)$  My attempt at a solution:  If two metrics are topologically equivalent, then they have the same convergent sequences. Honestly, I couldn't do anything. I am trying to define a sequence of functions $\{f_n\}_{n \in \mathbb N}$ such that $f_n \to f$ in, for instance, $(C^1([0,1]),d)$ but $f_n \not \to f$ in $(C^1([0,1]),d_{\infty})$. Could it be this two metrics are topologically equivalent? If this is the case, how could I prove it? If not, I would appreciate any hint to find an adequate sequence of functions that works for what I am trying to prove. Dynamic LINQ OrderBy on IEnumerable / IQueryable I found an example in the  for Dynamic LINQ that allows you to use a sql-like string (e.g. OrderBy("Name, Age DESC")) for ordering. Unfortunately, the method included only works on IQueryable&lt;T&gt;. Is there any way to get this functionality on IEnumerable&lt;T&gt;? Are there surface-mount pin header footprints for KiCAD? how to fill intersection of 2 shapes in latex tikz package I have to fill a specific region as given in the following figure.  The code I have written is   \begin{tikzpicture}  \draw (0,0) rectangle (10,10);  \draw (3.5,3.5) circle (5) node {A};  \end{tikzpicture}   Now, how to fill the intersecting area (area of circle overlapping the square) I have a 32-bit machine, can I use programs that are meant for 64-bit processors? I am using a 32-bit machine, but I am having to use a relatively obscure semantic database which only comes in 64-bit versions.  What are my options for running it in my local dev environment?  Currently when I try to run it, I get an error message that looks like this:  cannot execute binary file   and when I check my machine info, I get this:  $ uname -a Linux slabrams-desktop 2.6.32-29-generic #58-Ubuntu SMP Fri Feb 11 19:00:09 UTC 2011 i686 GNU/Linux Cannot Edit Python Script tool using its context menu from ArcMap/ArcCatalog? I am unable to edit scripts in ArcMap or ArcCatalog.  I navigate to the script, right-click on it and select "Edit", then nothing happens. I am using notepad++ as my editor (set in geoprocessing options). I have looked into this issue () and have attempted the steps therein. However I am still getting the same result, nothing happens.  I am using ArcGIS for Desktop Basic Version 10.3.1 and Python 2.6  What steps should I attempt to solve this issue?