I want to study the convergence of the series of general term $sin(\pi en!)$ In my naive way I see that this limit does not exist at infinity since e is an irrational number but they say it's zero Any explanation to this???? Thanks in advance
I didn't know how to solve this problem.  How to prove $$\sum_{n=0}^{\infty}\sin(e\pi n!)$$ converges?
Why does factoring eliminate a hole in the limit? $$\lim _{x\rightarrow 5}\frac{x^2-25}{x-5}  = \lim_{x\rightarrow 5} (x+5)$$  I understand that to evaluate a limit that has a zero ("hole") in the denominator we have to factor and cancel terms, and that the original limit is equal to the new and simplified limit. I understand how to do this procedurally, but I'd like to know why this works. I've only been told the methodology of expanding the $x^2-25$ into $(x-5)(x+5)$, but I don't just want to understand the methodology which my teacher tells me to "just memorize", I really want to know what's going on. I've read about factoring in abstract algebra, and about irreducible polynomials (just an example...), and I'd like to get a bigger picture of the abstract algebra in order to see why we factor the limit and why the simplified is equal to the original if it's missing the $(x-5)$, which has been cancelled. I don't want to just memorize things, I would really like to understand, but I've been told that this is "just how we do it" and that I should "practice to just memorize the procedure." I really want to understand this in abstract algebra terms, please elaborate. Thank you very much. How to add coloured line before caption? I have a bunch of files as follows:  04602635_b0294.DAT20120807164534 04602637_b0297.DAT20120807164713 04602638_b0296.DAT20120807164637 04602639_b0299.DAT20120807164819 04602640_b0298.DAT20120807164748 04602641_b0300.DAT20120807164849 04602650_b0301.DAT20120807164921 04602652_b0302.DAT20120807164956   I need to rename them to exclude the prefix. It needs to look like this..  b0294.DAT20120807164534 b0297.DAT20120807164713 b0296.DAT20120807164637 b0299.DAT20120807164819 b0298.DAT20120807164748 b0300.DAT20120807164849 b0301.DAT20120807164921 b0302.DAT20120807164956   EDIT  I forgot to add that I am using Solaris. Determine Which of the following Sets is a Vector Space Determine which of the following sets is a vector space.  $V$ is the line $y=x$, in the $xy$-plane: $V =\{[x,y]: \; y=x\}$  $W$ is the union of the first and second quadrants in the $xy$-plane: $W=\{[x,y]: \; y\geq0\}$  $U$ is the line $y = x+1$ in the $xy$-plane: $U=\{[x,y]: \; y=x+1\}$    I know that the answer is that "$V$" is the only vector space. However how do I come up with a quick proof to show that $V $is a vector space and that the other $2$ aren't? If you have an array of Java objects which have a primitive type (for example Byte, Integer, Char, etc).  Is there a neat way I can convert it into an array of the primitive type?  In particular can this be done without having to create a new array and loop through the contents.  So for example, given  Integer[] array   what is the neatest way to convert this into  int[] intArray   Unfortunately, this is something we have to do quite frequently when interfacing between Hibernate and some third party libraries over which we have no control.  It seems this would be a quite common operation so I would be surprised if there's no shortcut.  Thanks for your help! Why is it so difficult to find the twitter account for a Stack Exchange site? I am not able to find a link to the twitter account of Academia SE (), for example, in the Academia SE main webpage.   Why are twitter links not included on the main site page of each Q&amp;A site in SE? "All sites" overview should not require scrolling when selecting a site Django: How can I protect against concurrent modification of database entries He will understand that I was not joking.   He will understand I was not joking.  Which of the sentences is correct? Are there any specific rules about the use of &quot;that&quot; in the sentences I reported as example? Associates in Integral Domain Let $x$ and $y$ be nonzero elements of an integral domain $D$. Then $x$ and $y$ are associates if and only if $x = yd$ for some unit $d \in D$.    I am done proving the $\Leftarrow$ part. For $\Rightarrow$ what I did was: If $x$ and $y$ are associates then $x \mid y$ and $y \mid x$, so $x \mid y$ implies that $xs = y$ for some $s \in D$ and $y \mid x$ implies that $yt = x$ for some $t \in D$. That is, $yts=y$ which implies $ts=1$. Therefore $s$ is a unit and $t$ is a unit.    Am I on the right track? Can I say that the proof is complete? Use of definite article before phrases like Heathrow Airport, Hyde Park, Waterloo Station, Edgware Road and Parliament Square How can a question be tagged with a tag synonym? The question  seems to be tagged with , which, last I checked, is a synonym of . How is it possible that a question can be tagged with a synonym? Don't synonyms automatically get replaced on questions? Could anyone give me an explanation of the meaning of "get one's finger in the air" and the usage of it? Getting around the FAT32 4GB file size limit Taking advantage of one-time pad key reuse? Suppose Alice wants to send encryptions (under a one-time pad) of $m_1$ and $m_2$ to Bob over a public channel.  Alice and Bob have a shared key $k$; however, both messages are the same length as the key $k$.  Since Alice is extraordinary lazy (and doesn't know about stream ciphers), she decides to just reuse the key.  Alice sends ciphertexts $c_1 = m_1 \oplus k$ and $c_2 = m_2 \oplus k$ to Bob through a public channel.  Unfortunately, Eve intercepts both of these ciphertexts and calculates $c_1 \oplus c_2 = m_1 \oplus m_2$.     What can Eve do with $m_1 \oplus m_2$?     Intuitively, it makes sense that Alice and Bob would not want $m_1 \oplus m_2$ to fall into Eve's hands, but how exactly should Eve continue with her attack? Before upgrading to macOS Sierra, I really liked the Character Accent Menu by doing a long press on a key (for example, the character e) and being able to add an accent on it.   However, this functionality seems to not work anymore, and so far, I can't find out why?   Also, before, doing a long press on the delete button, would repeatly delete characters. Now, it is doing only one character deletion disregarding how long I'm pressing the key. So if I want to erase 10 characters, I need to press the delete button 10 times...  Also, I am not able to log off or shut down. When doing it, all applications are closed, status bar disappears but nothing happens. The dock remains here. I have no way to shut down or log out without doing a long press on the power button.  Is someone facing the same issue?  EDIT : Logging with a new user account or guest account fix all the problems. But I still would like to know what, in my user account, became corrupted during the upgrade. Prove that $\lim_{x \to \infty} \frac{\log(1+e^x)}{x} = 1$ Show that $$\lim_{x \to \infty} \frac{\log(1 + e^x)}{x} = 1$$ How do I prove this? Or how do we get this result? Here $\log$ is the natural logarithm. What is a catchy word that means (non-)self-descriptive I'm having fun with paradoxes, and I'm trying to come up with a catchy word for non-self-descriptive.  Maybe "autosemantic" fits the bill for self-descriptive (definitely open to other suggestions!), but I'm having a hard time negating it in a natural way.  For an example of the sentence I'm trying to construct:     Is "inautosemantic" inautosemantic?   It just isn't flowing. Photoelectric effect in a a frequency range I know how to calculate the kinetic energy of the emitted electrons   $$KE=hf-\phi$$  And the number of the emitted electrons, but this is valid for a certain frequency  $$N=\frac{IA}{hf}=\frac{P}{hf}$$  What happens if I have a frequency range? for example between λ1 and λ2?  BTW: It's not homework, I'm an engineering student,  not a physics student Let $f$ be a continuous and open map from $\mathbb R $ to $\mathbb R$.Prove that $f$ is monotonic. Let $f$ be  a continuous and open map from $\mathbb R $ to $\mathbb R$.Prove that $f$ is monotonic.  Suppose that $f$ is not monotonic.Then $\exists a,b;a&gt;b$ such that $f(a)&lt;f(b)$ and $c,d;c&lt;d$ such that $f(c)&gt;f(d)$.Since $f$ is continuous then there exists no break in the graph of $f$. But how should I use the fact that $f$ is open.Any help on how should I do the proof?