Situation 1:  A regular $n$-gon is inscribed in a circle. As $n$ increases without bound, the area of the $n$-gon approaches the area of the circle and the perimeter of the $n$-gon approaches the circumference of the circle.  Situation 2: Consider a $1$ by $1$ square with one side labeled South and the other labeled North East and West as in a map.  A path is constructed from the Southwest corner to the Northeast corner.  If the path runs east on the south side for a distance $\frac{1}{2^n}$, then goes north for the same distance, then east again for distance $\frac{1}{2^n}$. And so on.  Then the total length of the path is $2$  As $n$ increases without bound the area under the path and above the south side of the square approaches the area under the diagonal, but the length of the path remains $2$ and does not approach the length of the diagonal.  Why is there a difference?
What is wrong with this proof?    Is $\pi=4?$
How to call a SOAP web service on Android I am having a lot of trouble finding good information on how to call a standard SOAP/WSDL web service with Android. All I've been able to find are either very convoluted documents and references to "kSoap2" and then some bit about parsing it all manually with . OK, that's fine, but it's 2008, so I figured there should be some good library for calling standard web services.  The web service is just basically one created in . I would like to have IDE support for generating the plumbing classes. I just need the easiest/most-elegant way to contact a  based web service from an Android-based phone. Voting on Meta sites generally express agreement and disagreement with a proposal, unlike on the Q&amp;A sites where they indicate useful/good vs. non-useful/bad content.  It seems to me that a down-vote on Meta Stack Exchange should not carry a -1 rep penalty, which only discourages down-voting. With this discouragement, questions on Meta SE will be biased towards agreement, since for every up-voter, there may be several in disagreement who don't want to lose the rep for down-voting.  Can the penalty for down-voting on Meta SE be removed? I seem to remember from first year physics that we can calculate the RMS speed of a stationary, ideal gas with $v=\sqrt{\frac{3RT}{M}}$. Does a similar equation exist for liquids? I can't figure how to print the string "-n" with the over 30 years-old command echo.  I tried these:  Michaels:Scripts$ echo -n Michaels:Scripts$ echo "-n" Michaels:Scripts$ echo -e "-n" Michaels:Scripts$ echo -- -n -- -n Michaels:Script$ echo -- "-n" -- -n Michaels:Script$ echo "\-n" \-n   I use Mac OS X ML. I consider this behaviour very weird and unexpected. 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But what causes a light wave (or an electromagnetic wave) to fade if there are no celestial bodies in its way? Or, alternatively, why doesn't it fade sooner? Some stars are visible to the naked eye that are hundreds of light years away. Others, closer, can only be detected with the aid of a telescope. Why? Area 51 login no longer redirects nor logs in Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ having the following properties?   $f(x) = 0$ for all $x \le 0$. $f(x) = 1$ for all $x \ge 1$. For $0 &lt; x &lt; 1$, $f$ is strictly increasing. $f$ is everywhere $C^\infty$. The sequence of $L^\infty$ norms $\langle \left\lVert f \right\rVert_\infty, \left\lVert f' \right\rVert_\infty, \left\lVert f'' \right\rVert_\infty, \dots \rangle$ is bounded.   If we impose only the first four conditions, there is : for $0 &lt; x &lt; 1$, define $f(x)$ by $$ f(x) = \frac{e^{-1/x}}{e^{-1/x} + e^{-1/(1-x)}} = \frac{1}{1 + e^{1/x - 1/(1-x)}} $$ However, the derivatives of this function appear to grow quite rapidly. (I'm not sure how to verify this, but it seems at least exponential to me.)  If such a function does not exist, what is the smallest order of asymptotic growth that the sequence $\langle \left\lVert f \right\rVert_\infty, \left\lVert f' \right\rVert_\infty, \left\lVert f'' \right\rVert_\infty, \dots \rangle$ can have? Can't delete folder in Windows 7 How to avoid duplicating assigned materials when duplicating a mesh? how can I prevent Blender 2.8 from duplicating materials every time (?) I duplicate a mesh. Error sending edittext in java server. See code