I think the equation above is false, but I cannot find a counterexample.  Also, I think that the implication is true when $a_n^2 \to 1$ is also required.  Suppose we have $a_n\in\mathbb{C}\ \forall n\in\mathbb{N}$.  Can you help me?
I am looking for an example of a sequence which shows that, $a_{n+1} - a_n \to 0$ as $n \to \infty$ does not imply that sequence $a_n$ converges. I have a feeling that the sequence should be an oscillatory one but I am unable to think of an example.
What is reflection, and why is it useful?  I'm particularly interested in Java, but I assume the principles are the same in any language. Suppose we have a natural number $n \ge 0$.  Given natural numbers $\alpha_1,\ldots,\alpha_k$ such that   $k\le n$ $\sum_i \alpha_i = n$   what is the maximum value that $\Pi_i \alpha_i$ can take?  I'm quite sure that there is a theorem telling me the result, but I cannot find it. For sure an upper bound is $n^k$ but I'm searching for a real upper bound. I'm pretty sure that upper the bound should be $n^2$, but I don't know I could prove it. Using 3rd party Python modules I'm currently developing a script for Blender to handle Mesh Frequency Decomposition.  The script is nearly complete, but i need one more thing: The SciPy library to compute eigenvalues and eigenvectors.  Following the installation instructions on , I'm able to install it as well as Numpy. Numpy works fine, but I can't even import SciPy:  Traceback (most recent call last): File "/CAF_FD.py", line 4, in &lt;module&gt; ImportError: No module named 'scipy' Error: Python script fail, look in the console for now...   By the way, I'm working on Linux 64 bit and I've installed SciPy and Numpy with my package manager.  Edit: When I try to use SciPy with a terminal prompt, it works. CSS selector for first element with class Why doesn't the Related question list show that questions are closed? Change Detection using Raster Calculator I'm currently analyzing the change detection for urban expansion over a decade starting from 2006-2016. How can I use raster calculator in order to know the changes? For example, from agriculture to urban area and etc. I already classified the images into 4 classes, urban area,vegetation,water bodies and barren land. Piping data through arbitrary functions in Clojure Evaluating the infinite product $\prod\limits_{k=2}^\infty \left ( 1-\frac1{k^2}\right)$ Evaluate the infinite product $$\lim_{ n\rightarrow\infty }\prod_{k=2}^{n}\left ( 1-\frac{1}{k^2} \right ).$$  I can't see anything in this limit , so help me please. How do I determine the size of my array in C? This might be a duplicate of  but that question is 5 years old and unanswered.  When I'm in drive.google.com and I open a G Suite file (either double-click or select &amp; press enter) in Firefox it opens in a new window whereas in Chrome it opens in a new tab.   Why do they do that? Is this a browser-specific quirk or a Google setting (that I haven't found)? Is there a way to control the behavior? I'd like them both to open in tabs, but presumably the same answer would be relevant in the opposite direction too. MLE for Uniform $(0,\theta)$ I am a bit confused about the derivation of MLE of Uniform$(0,\theta)$.  I understand that $L(\theta)={\theta}^{-n}$ is a decreasing function and to find the MLE we want to maximize the likelihood function.  What is confusing me is that if a function is decreasing, then wouldn't the function be maximized at the smallest input rather than the largest?  Thank you in advance for your help. This is from  .      Let $\mathcal{A}$ be a locally small category, and let $A,A' \in \mathcal{A}$ with $\operatorname{Hom}(-,A) \cong \operatorname{Hom}(-,A')$. Prove directly that $A \cong A'$.     My thoughts:      Let $\eta$ denote the natural isomoprhism between the two functors.    We have    $$H(A,A) \xrightarrow{ \eta_A} H(A,A')$$   So $\eta_A(id_A): A \rightarrow A'$ is a candidate. Now I want to construct inverse.    $$ H(A',A') \xrightarrow{\eta_{A'}} H(A',A)$$   Then the map $\eta^{-1}_{A'} (id_{A'}):A' \rightarrow A$.    So I'd like to show that the composition is identity. By the naturality condition of $\eta$, from $H_A(A) \rightarrow H_{A'}(A')$ (and similarly on other direction)    $$ \eta_A(id_A) \circ \eta^{-1}_{A'}(id_{A'}) = \eta_{A'}(\eta_{A'}^{-1}(id_{A'}))= id_{A'} $$   We deduce these two maps are inverses. Hence $A' \cong A$.    Is this correct? Or is there a neat way to see this? let's say that I have a following file structure:  ~/DIR1 -fileA -DIR2 --fileB   Then, being in ~, I run this script:  cp DIR1 NEWDIR2   And just an error happens:  cp: omitting directory ‘DIR1’   I understand why nothing from DIR1 gets copied (because I didn't use -r switch that would enforce recursion) but I'd expect the DIR1 folder itself copied to NEWDIR2 (so practically a creation of a new "NEWDIR2" in ~ or a creation of ~/NEWDIR2/DIR1).  Could someone explain to me why is my expectation wrong? Reading Chapter 2 of Koblitz's Introduction to Elliptic Curves and Modular Forms, I got stuck on the following question. I would like to proceed my reading, so I would appreciate any hint to this. I have no background in Algebraic Number Theory, but if someone explains in that language, I would like to see it and learn it as much as I can at this moment. References will also be appreciated!  Let $q$ be a power of prime $p$ such that $q \equiv 1 \mod 4$ and $p \nmid 2n$.  Question. Let $\chi_{4} : \mathbb{F}_{q}^{*} \rightarrow S^{1}$ be the character of order $4$, show that $\chi_{4}(-4) = 1$.  The hint in the book says we may show that $\chi_{4}(4) = \chi_{4}(-1) = 1$ for $q \equiv 1 \mod 8$ and $\chi_{4}(4) = \chi_{4}(-1) = -1$ for $q \equiv 5 \mod 8$.  If $q \equiv 1 \mod 8$, then $\chi_{4}(4) = \chi_{4}(2)^{2} = \chi_{2}(2) = \pm 1$, where $\chi_{j}$ denotes the character of order $j$. So I think $2$ must be quadratic residue in this case, but I cannot figure out why. To make your opinion known on MSE you need both up and downvotes. Given at the moment you need 125 rep to downvote on this site, the voting patterns will be skewed positive because not everyone is active on the site.   This is to be expected for regular Stack Exchange sites as you are expected to be involved in the community before you are deemed knowledgeable enough to be part of the community voice. However, MSE is different because you are voting for the entirety of the Stack Exchange network and you shouldn't need to be active on MSE itself to have a voice on changes to the entire network.  This is especially important for questions like  because it will effect every network to do with coding at all. This was brought up in .  I would suggest reducing the rep required to downvote on MSE down to 100 (or 101) rep. This is the amount you get for the site association bonus. This is given once you achieve 200 rep on any network site and shows that you are active in at least one community. I think this would be the perfect use of the association bonus in showing that you are a member of the Stack Exchange community. I've been reading around and trying to figure out how to do this, but for some reason I can't seem to override parent functions in my child theme.  I'm using TwentyTen as a parent - can anyone tell me why this function in my child theme isn't overriding the parent function please?  // Override read more link function osu_twentyten_continue_reading_link() {  return ' &lt;a href="'. get_permalink() . '"&gt;' . __( 'Read on &lt;span class="meta-nav"&gt;&amp;rarr;&lt;/span&gt;', 'twentyten-child' ) . '&lt;/a&gt;'; } function osu_twentyten_auto_excerpt_more( $more ) {  return ' &amp;hellip;' . osu_twentyten_continue_reading_link(); } remove_filter( 'excerpt_more', 'twentyten_auto_excerpt_more' ); add_filter( 'excerpt_more', 'osu_twentyten_auto_excerpt_more' );   I thought you had to remove the filter/action etc. before re-adding it right?  Thanks,  osu Are the LIGO observations proof of a black hole merger, and what happened to the black holes? Recent reports claim that the gravitational waves detected by LIGO match up with the signal expected from two black holes merging as predicted by general relativity.  Additionally, the masses of both black holes were estimated. Questions:  How certain are we that the detected of gravitational waves are necessarily from two black holes merging?  Since supermassive black holes are typically at the center of a host galaxy, what happened to the galaxies that contained the two merging black holes?  For example, did their host galaxies merge?    Edit After reading the answers, I understand that the signals detected by LIGO are proof of the validity of gravitational wave generational artifacts as predicted by general relativity.  When additionally gravitational waves, caused by the merger of the supermassive black holes (those which ARE indeed located in the centers of the merging galaxies) will be detected, since mergers of galaxies could be observed by the methods independent of gravitational waves such as registering the fact of the quasar appearance (). Additional questions:  Could someone offer me the historical ASTRONOMY precedent when the second degree of inference (from the mathematical model, being associated with the theory of Physics) was accepted as the discovery of the astronomical object?  Could someone give a reference to trustable scientific publication, with the subject of study being discussion of requirements satisfying claim of astronomical discovery of an astronomical object?  Have independent scientific sources (outside of members of the LIGO team) analyzed methodology and results of the LIGO signal detection, and published their conclusions with regards of what was actually discovered? Compute integral $\int_{-\infty}^\infty \frac{x^4}{1+x^8} \, dx$ Compute the integral.   $$\int_{-\infty}^\infty \frac{x^4}{1+x^8} \, dx$$  The answer at the back of the book is  $$\frac{\pi}{4\sin(\frac{3\pi}{8})}$$ If/When to move to secondary fermentation Digression from   What are the good rules of thumb for determining a. whether to use secondary fermentation at all and b. when to move from primary to secondary.  Preferably any rule of thumb should have both chemical and anecdotal rationale supporting it. Internal or external oscillator I always use the internal oscillator that pics have as I have never found the need to run anything at higher frequency than 8 MHz (which is the fastest the pics I use tend to be able to go). Are there any reasons, beyond going above 8 MHz, that mean I should use an external oscillator? It just seems like one more thing to go wrong to me but I'd be interested to hear what others do.