show $ \ln(1+\frac{1}{x})−\frac{1}{x+1}&gt;0 $ $;$ $\forall x&gt;1$. I tried to show $ \frac{1}{2x^2} - \frac{2}{3x^3} + \frac{3}{4x^4} - \frac{4}{5x^5} \pm ... &gt;0$ $;$ $\forall x &gt;  1$ . help me.
$$\frac{1}{n+1}&lt; \log(1+ 1/n)$$  Any ideas? I tried estimating the difference between $1/n$ and the logarithm and comparing with $1/n-1/(n+1)$ but I miserably failed.
She don't know that she's a lucky girl Here's from an old song (1963) "" by Lesley Gore.       She don't know that she's a lucky girl   Here's another example, "Silence is Golden" (1964) by The Tremeloes [YouTube ]      Oh, don't it hurt deep inside to see someone do something to her.   I wonder how common it is to say "She don't" or "don't it". Do native English speakers often use these?  By the way, Lesley Gore is American, and the Tremeloes is British. How can a file be copied? Assuming I have an ArrayList  ArrayList&lt;MyClass&gt; myList;   And I want to call toArray, is there a performance reason to use  MyClass[] arr = myList.toArray(new MyClass[myList.size()]);   over  MyClass[] arr = myList.toArray(new MyClass[0]);   ?  I prefer the second style, since it's less verbose, and I assumed that the compiler will make sure the empty array doesn't really get created, but I've been wondering if that's true.  Of course, in 99% of the cases it doesn't make a difference one way or the other, but I'd like to keep a consistent style between my normal code and my optimized inner loops... 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.) I'm wondering whether I should use the present tense for a descriptive statement but in a past context. For example:     (1) I went to the zoo and saw a giraffe last week.       (2) A giraffe is a mammal with long neck.       (3) It is the first time I see that animal with my own eyes.   Statement (2) is a description of a giraffe in general, not the specific one in the zoo, so is it correct to use is?  Statement (3) is an event that is still true now, so should I use is and see?  I'm not native speaker so forgive me if the question is very obvious to others. How do I replace multiple spaces with a single space in C#? How can I replace multiple spaces in a string with only one space in C#?  Example:  1 2 3  4    5   would be:  1 2 3 4 5 I only read part of the second book in this series years ago, and am unsure of how many books were in it. It was about a teenage or young adult boy, who was part of the ruling class of this fantasy world. He was gay, and his lover was the heir to the throne. The antagonist of the first book - I haven't read it, but I got the gist from comments in the second - was the current queen, who tricked the boy and his lover into leaving the kingdom for remote outer regions, thus enabling her son to take the throne. She then married her own son to keep herself in power.  The second book had the boy and his lover slowly becoming enemies, as the lover began to corrupt the simple tribesmen they were sheltering with in his quest to regain his throne. He wanted to build an empire in the hinterland and use the army to attack his homeland, killing the queen in the process, and didn't care about the long-term effects his actions were having on the culture or environment of his new followers.   The lover defeats a tribesman by breaking his neck in unarmed combat, then later takes a young boy as his "willing catamite." He also leads raids on neighbouring tribes, establishes alliances with other neighbouring tribes and organises daring and successful raids on the eggs of dinosaurs living in the middle of a swamp. He enslaves the first tribe to turn against him, and gives their children to slavers from the empire in place of those of allied tribes, thus weakening his enemies while ensuring the loyalty of his allies.  The series involved dinosaurs living among humans, being both domesticated and wild, used as livestock, for hunting and as transport. The boy and his lover were much taller than the average humans who were forced to pay tribute to this pseudo-magical empire they came from; all the humans from the empire were much larger than their subjects. The tribesmen have a tradition about not touching the ground with bare feet, as it is an insult to nature, or their god. The boy and his lover both ignore this superstition, but the catamite also begins to ignore it, despite looking ashamed as he does so.  I have absolutely no idea when this series was written, what it is called or who wrote it. All I know is that I read it in 2005-06, though it probably wasn't new. I'm hoping the long description I've given will help someone work it out. Is there a way to see what were the Candidate Execution Plans generated by the Query Optimizer? The Query Optimizer creates more than one possible execution plan. How can I see all the plans generated before the executed one was chosen?   :     The Query Optimizer must analyze the possible plans and choose the one   with the lowest estimated cost.   Please, notice that I'm not talking about , I'm talking about candidate execution plans, the ones that were generated but not selected to be executed.  According to Benjamin Nevarez's article :     The generation of candidate execution plans is performed inside the   Query Optimizer using transformation rules, and the use of heuristics   limits the number of choices considered in order to keep the   optimization time reasonable. Candidate plans are stored in memory   during the optimization, in a component called the Memo.   Would it be possible to visualize the candidate plans rendered the way we can ? Convert web page to image $$f(x,y) = \left\{\begin{array}{cc} \frac{xy}{x^2+y^2} &amp; (x,y)\neq(0,0) \\ f(x,y) = 0 &amp; (x,y)=(0,0) \end{array}\right.$$  In order to verify if this function is differentiable, I tried to prove it by the theorem that says that if $\frac{∂f}{∂x}$ and $\frac{∂f}{∂y}$ exist and are continuous at the point $(x_0,y_0)$, then the function is differentiable at this point. So I did:  $$\frac{\partial f}{\partial x}(0,0) = \lim_{h\to 0}\frac{f(0+h,0)-f(0,0)}{h} = 0$$ $$\frac{\partial f}{\partial y}(0,0) = \lim_{h\to 0}\frac{f(0,0+h)-f(0,0)}{h} = 0$$  so we have that the partial derivatives at point $(0,0)$ is $0$. Now, if we take the derivative at $(x,y)\neq (0,0)$ and then take the limit of it as $(x,y)\to(0,0)$, we can see if the derivatives are continuous or not. So here it is:  $$\frac{\partial f}{\partial x}(x,y) = \frac{y(y^2-x^2)}{(x^2+y^2)}$$  but   $$\lim_{(x,y)\to(0,0)} \frac{y(y^2-x^2)}{(x^2+y^2)} $$ does not exist (by wolfram alpha... but can anybody tell me an easy way to prove this limit does not exist? easier than taking the limit in different directions?), therefore the derivative is not continuous at $(0,0)$, so we can't say $f$ is differentiable at $(0,0)$, but for $(x,y)\neq (0,0)$ the function is continuous, as it is a quotient of continuous functions. So $f$ is at least differentiable at $(x,y)\neq (0,0)$.   Now, to verify differentiability at $(0,0)$ I think we must use the limit definition of differentiablity: A function is differentiable at $(0,0)$ iff: $$\lim_{(h,k)\to (0,0)} \frac{f(0+h,0+k)-f(0,0)-\frac{\partial f}{\partial x}(0,0)-\frac{\partial f}{\partial y}(0,0)}{\|(h,k)\|} = 0$$  Let's calculate this limit:  $$\lim_{(h,k)\to (0,0)} \frac{f(0+h,0+k)-f(0,0)-\frac{\partial f}{\partial x}(0,0)-\frac{\partial f}{\partial y}(0,0)}{\|(h,k)\|} = \\ \lim_{(h,k)\to (0,0)} \frac{\frac{hk}{h^2+k^2}}{\sqrt{h^2+k^2}} = \\ \lim_{(h,k)\to (0,0)} \frac{hk}{(h^2+k^2)\sqrt{h^2+k^2}}$$  which I think, it's a limit that does not exist, therefore the function isn't differentiable at $(0,0)$ In a simple battery + resistor circuit, what form of energy is lost from the electrons upon exiting the resistor? I will give this question a little context. Firstly, as I understand it, as soon as I "close the switch" on a circuit, electric current pretty quickly establishes a steady state where, at any given cross section along a wire, the average kinetic energy of that slice is both constant and equal anywhere along the wire. (Please correct me if I am misunderstanding).   If the average kinetic energy is constant anywhere throughout the wire in the circuit (which confuses me, as I would think that the electrons should all accelerate after exiting the resistor since they are no longer being impeded as much), that means that the average velocity is constant. So, in the example of a simple circuit with a battery of v Volts and a resistor of o Ohms, my question is the following:  Because a 'voltage drop' is known to occur across the resistor, what type energy is being "traded" to generate the heat that radiates from the resistor? I would think that the reflexive answer is "electrical energy"...hence the VOLTAGE drop (the sacrificed energy is clearly not kinetic, as the velocity is constant everywhere). However, I find this confusing. Does this mean that if I had a positive test charge, it would be easier for me to bring it to the beginning of the resistor as compared to bringing it to the end of the resister?     Further, if the correct answer IS "electrical energy", why exactly DOES electrical energy get "dissipated" as the electrons pass through the resistor? I always here the comment Oh! It's because all the electrons are running into densely packed crap which impedes their flow but that to me sounds like a reason for their KINETIC energy to be reduced. However, clearly that's not the case. So, ultimately, I guess the real question is:  **What goes on in the resistor that is literally removing electrical energy **. I am sure that this is a quantum mechanics question, but my quantum mechanics knowledge is not terribly strong. If I could get an answer that is devoid of crazy wave function equations, I would greatly appreciate it. Thanks! Mirror cells so that a change in either cell updates the other I have 2 cells containing the same picklist values on separate sheets. I want the cells to mirror each other so that whenever I choose a new value in either sheet 1 or sheet 2, the value on the other sheet is updated to match the newly selected value. Is this possible?  UPDATE   Worksheet 1:   Worksheet Name: 'Live Forecast Tool' Cell containing picklist: C3:D3 (merged) Picklist values within range: ='SF MRF'!H5:Y5  Worksheet 2:   Worksheet Name: 'AH' Cell containing picklist: C3:D3 (merged) Picklist values within range: ='SF MRF'!H5:Y5    Cell C3:D3 needs to update to show the value that has been changed on the other sheet. Possible Duplicate:          I have seen  in SQL. I wonder how would you approach the opposite problem: splitting coma delimited string into rows of data:  Lets say I have tables:  userTypedTags(userID,commaSeparatedTags) 'one entry per user tags(tagID,name)   And want to insert data into table  userTag(userID,tagID) 'multiple entries per user   Inspired by  question  EDIT  Thanks for the answers, actually more then one deserves to be accepted but I can only pick one, and the  with recursions seems pretty clean to me. It works on SQL Server 2005 and above.   For earlier version of SQL Server the solution  can be used.  For working with text data type  will be helpful. Thanks again. How can I delete my account? How can I delete my Stack Exchange account(s)? Also,  What happens to my content? Can I request it be deleted as well? What if I'm suspended, rate-limited, or banned from posting questions or answers? What will happen to my votes? Will other users be impacted? Why wasn't my account deleted immediately after the timer expiring? What will happen to my pending bounties and close, reopen, delete, or undelete votes? I want my account deleted on all Stack Exchange sites. How do I do this? Will my profile on chat be removed once I delete my site account?   For more information, see &quot;&quot; in the . Why does hair grow after trimming but remains at a constant length after a while? Some hair especially body hair regrows after trimming but stops growing after a while. What is the mechanism behind control of hair growth and how is the length limit determined? Extra "lines" are being added when creating/editing questions/answers in Edge Lately, I've noticed that extra lines are being added when I create questions/answers. For example, if I write:   Your problem is X, Y, Z. Fix it by doing:      Code  Then maybe also do some other thing.  Oh, and some other thing as well.   When I post it, then go back into the editor, it's been changed to:   Your problem is X, Y, Z. Fix it by doing:       Code   Then maybe also do some other thing.   Oh, and some other thing as well.   Which is mildly annoying since that causes less to be displayed in the box at once, but whatever.  The weird thing is though, the added line isn't properly treated as a newline, even though it does appear to be caused by a newline. A lot of lines that look like they should be empty don't even get rendered. If I delete the original lines that I entered leaving only the "new" newlines, I get weird behavior:    I tried selecting the characters involved and inspecting them in Python, and they just appear to be newlines, so I have no idea what the problem is.    The only browser I've been able to reproduce it in is Edge (version 44.18362.329.0). It does not happen in Chrome. This has been happening for ~3 days now.    There is also a problem with link embedding and code formatting. If while editing I highlight some text and either try to embed a link (interlinked circle button) or format it as code ({} button), it often puts the wrong text between the […]s when embedding, or formats code that I didn't highlight (and doesn't format the code that I did highlight). Do the same camera settings lead to the same exposure across different sensor sizes? Let's say I have a micro-4/3rd camera and a full frame camera, both set to 1/60 at f/2.8, taking a picture of the same scene in the same lighting. Will the exposure be the same across both cameras despite the different sensor sizes?  The reason why I'm asking is because of the difference in depth of field between micro-4/3 and full frame sensors. I'm finding that, in order to take a picture of certain scenes with the full frame camera at the same depth of field as the micro-4/3rd camera, I have to increase the aperture, which in turn forces me to crank up the ISO. How to stop Gedit, Gvim, Vim, Nano from adding End-of-File newline char? How to set the command prompt's color in Bash? State-of-the-art algorithms for the training of neural networks with GRU or LSTM units I recently read a lot about neural networks using GRU or LSTM units. There are many easy to use frameworks like tensorflow that do not even require high knowledge about programming. Unfortunately, I never really found good information on how the training of those networks work. Simple backpropagation might probably not work for gated recurrent networks or is just too inefficient for networks with such a high number of variables to learn.  So my question is:  What are the state-of-the-art algorithms used for the initialization and training of neural networks with GRU or LSTM units? I am not looking for frameworks to use, but for initialization and update equations for the internal parameters.