Does $S = \sum_{n=1}^{\infty}(-1)^n (e- (1+ \frac{1}{n})^n)$  converge or diverge?  My attempt  : I know that  $e = \lim_{n\rightarrow \infty}( 1+ \frac{1}{n})^n$.  Now put the value  e in   given series $S$ , I got $\sum_{n=1}^{\infty}(-1)^n (e- (1+ \frac{1}{n})^n)= \sum_{n=1}^{\infty}(-1)^n (e- e)=0$  so the given series  is converges    is  it  correct????
I am having a lot of trouble with these series questions. Up until this point, I had relatively little trouble with all the questions in the book. These seem to require knowledge about approximations of functions and other external experience-based knowledge, which I just don't have yet.  Determine convergence or divergence of the given series. In the case of convergence, determine whether the series converges absolutely or conditionally.  $$\sum_{n=1}^\infty (-1)^n\left[e-\left(1+\frac 1 n \right)^n\right]$$  It's easy to see that  $$\lim_{n\to\infty}\left[e-\left(1+\frac 1 n\right)^n\right]=0$$  however, in order to apply Leibniz's Rule and show conditional convergence I need to show that the sequence is monotonically decreasing. This doesn't seem doable with straight inequalities, so I tried taking the derivative, which just resulted in an uninterpretable mess. This doesn't even begin to address the question of absolute convergence/divergence.  There are 54 of these questions... I must be missing something really fundamental if they all take this long.
How to add citation for a whole paragraph? If I write a report and a whole paragraph is based on one source but not cited verbally it seems unnecessary to include the reference after every sentence:  Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua (source 1). Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat (source 1).  Now I found a Phd thesis that puts the reference after the last dot of the paragraph to indicate that it is based on this very source:  Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. (source 1)  Is this a common practice? Is it generally understood? What are other ways to add the reference for a whole paragraph? While using the Pixel Inspector, whenever I click "copy all values" it only copies values that you can see on the screen. Anyone know a solution? I need all the values to count the number of times 1, 2, and 0 appear on the graph. What are these circular splotches on my negatives? What does the typical workflow of a journal look like? What steps does a manuscript typically go through from submission to publication (or rejection) in a typical journal? How are these steps referred to, in particular by editorial systems, and how long do they each typically take?  Note that this question is about the typical situation and hence not about:   Journals with an atypical workflow, e.g. those that allow for an instantaneous reviewer–author interaction. Exceptional steps or rare occurrences such as withdrawal or .   This is a canonical question on this topic as per . Due to its nature, it is rather broad and not exemplary for a regular question on this site. Please feel free to improve this question. Sum with binomial coefficients: $\sum_{k=1}^m  \frac{1}{k}{m \choose k} $ `du -b` shows different size than `ls` Square bracket notation for dimensions and units: usage and conventions One of the most useful tools in  is the use of square brackets around some physical quantity $q$ to denote its dimension as  $$[q].$$  However, the precise meaning of this symbol varies from source to source; there are a few possible interpretations and few strict guidelines. What conventions are there, who uses them, and when am I obliged to follow them? Evaluation of Indefinite Integral resulting in Hypergeometric Function I am attempting to derive the result:  $$ \int \left(1+x^n\right)^{-1/m}dx= x\,_2F_1\left(\frac 1m,\frac 1n;1+\frac 1n;-x^n\right)$$  First, I start off with the binomial expansion of the integrand to get:  $$\int\sum_{k=0}^{\infty}\frac{(1/m)_k}{k!}\left(-x^n\right)^kdx  $$  Then, I pull out a $-1$ and interchange summation and integration:  $$\sum_k\int (-1)^k\frac{(1/m)_k}{k!}x^{nk}dx  $$  $$ \sum_k (-1)^k\frac{(1/m)_k}{k!}\frac{x^{nk+1}}{nk+1} $$  $$ x\sum_k \frac{(1/m)_k}{k!}\frac{(-x^n)^k}{nk+1}  $$  So, I'm just right there, but not sure how to express this series in the general form of a hypergeometric series with the two additional Pochhammer symbols.  I even typed this last series into Mathematica, and it returned the Hyper. Function. What am I missing, or how do I transform this last series into the desired form? If $|G| = pq$,  $p,q$ primes, $p \gt q, q \nmid p-1 $, then how do I prove $G$ is cyclic without using Sylow's theorems? Real Roots and Differentiation Prove that the equation $x^5 − 1102x^4 − 2015x = 0$ has at least three real roots.  So do I sub in values of negative and positive values of $x$ to show that there are at least three real roots? The method to do this question is not by finding the factors of $x$ right? Because it will be too tedious so I want to ask whats the other solution to prove this? Help appreciated. Thank you very much. How to prove $\sum\limits_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$? Suppose I've just made the switch to . I have chosen one of the built-in styles (numeric, alphabetic, authoryear, authortitle, verbose) or of the custom styles available at CTAN, and I've figured out how to sort/compress citations. The bibliography and the in-text-citations look approximately like I want, but need to be tweaked. I've taken a look at the biblatex documentation for advice, but it is somewhat daunting. How do I get started? To add some focus, customization in the following areas would come in handy:   Suppressing of specific BibTeX fields (e.g., publisher); Formatting of fields for specific entry types (e.g., titles of articles not inside quotation marks); Formatting of citation postnotes (i.e., getting rid of "p."/"pp."); Spacing and punctuation in the bibliography (e.g., commas instead of periods as unit separators); Order of first names and last names of authors/editors in the bibliography; Order of certain fields in the bibliography (e.g., swapping location and publisher); Language-specific terms (e.g., replacing "References" with "Works Cited").   Finally, assuming that I'll want to use my style adjustments in a multitude of documents: Where do I put the relevant macro stuff? What is the intuition behind beta distribution? Disclaimer: I'm not a statistician but a software engineer. Most of my knowledge in statistics comes from self-education, thus I still have many gaps in understanding concepts that may seem trivial for other people here. So I would be very thankful if answers included less specific terms and more explanation. Imagine that you are talking to your grandma :)  I'm trying to grasp the nature of beta distribution – what it should be used for and how to interpret it in each case. If we were talking about, say, normal distribution, one could describe it as arrival time of a train: most frequently it arrives just in time, a bit less frequently it is 1 minute earlier or 1 minute late and very rarely it arrives with difference of 20 minutes from the mean. Uniform distribution describes, in particular, chance of each ticket in lottery. Binomial distribution may be described with coin flips and so on. But is there such intuitive explanation of beta distribution?    Let's say, $\alpha=.99$ and $\beta=.5$. Beta distribution $B(\alpha, \beta)$ in this case looks like this (generated in R):     But what does it actually mean? Y-axis is obviously a probability density, but what is on the X-axis?   I would highly appreciate any explanation, either with this example or any other. I'm trying to compute this integral, $$\int_{0}^{\infty}\frac{\ln x}{x^{a}(x+1)}dx \hbox{ where } 0&lt;a&lt;1$$ I drew a typical Pacman contour with branch cut at positive real axis. Then, we have $$\int_{\Gamma}\frac{\ln z}{z^{a}(z+1)}dz=\left(\int_{L_{1}}+\int_{L_{2}}+\int_{C_{R}}+\int_{C_{\epsilon}}\right)\frac{\ln z}{z^{a}(z+1)}dz$$ The function has a pole at $z=-1$. Then, by residue theorem, the whole integral becomes  \begin{align*} \int_{\Gamma}\frac{\ln z}{z^{a}(z+1)}dz&amp;=2\pi i\text{res}(\frac{\ln z}{z^{a}(z+1)};-1)\\&amp;=2\pi i\lim_{z\rightarrow-1}(z+1)\frac{\ln z}{z^{a}(z+1)}\\&amp;=2\pi i\lim_{z\rightarrow-1}\frac{\ln z}{z^{a}}\\&amp;=2\pi i\frac{\ln e^{i\pi}}{e^{i\pi a}}\\&amp;=2\pi i\frac{\ln1+i\pi}{e^{i\pi a}}\\&amp;=-\frac{2\pi^{2}}{(e^{i\pi})^{a}} \end{align*} On $L_1$, $z=xe^{i\epsilon}$ where $\epsilon\leq x\leq R$  On $L_{2}$, $z=xe^{i(2\pi-\epsilon)}$ where $\epsilon\leq x\leq R$  On $C_{R}$, $z=Re^{i\theta}$ where $\epsilon\leq\theta\leq2\pi-\epsilon$  Lastly on On $C_{\epsilon}$, $z=\epsilon e^{i\theta}$ where $\epsilon\leq\theta\leq2\pi-\epsilon$  So I tried to compute them separately and I anticipate the curves will both go to $0$ by ML inequality, and two linear integrals will consist of the term $\int_{0}^{\infty}\frac{\ln x}{x^{a}(x+1)}dx$ and hence simply solve for it. But I'm having trouble physically solving it out. I think the log is throwing me off. Can anybody please help me? In bibliography, how to print the page in my text where I cited certain paper? How to get, for instance,  [19] S. Roth and M. J. Black. Fields of experts: A framework for learning image priors. In IEEE Conf. on Computer Vision and Pattern Recognition, 2005. 1, 2, 7, 8  where the last numbers (1, 2, 7, 8) are the pages where I cited the paper in MY TEXT?  This would be like a reversed reference, or a index for bibliography. Why can't I move a window to the top left corner with wmctrl in Unity? I've been playing around with wmctrl to take more control of my windows. I've encountered a weird behaviour where Unity reserves double the space. See this screen shot:    I opened the terminal and moved the window using the top command. It should be at the top left corner (X=0 Y=0). But for some reason there is some margin between the terminal and the Unity bars.  If I understand the man pages correctly it might have something to do with the WA: 65,24.  Can anybody help me to fix this? Edit answer draft is broken The Edit Answer Draft button is broken.    Press Back with text in an answer box. The Edit answer draft bar does not appear.     Seems like an onBackPressed() issue.  Also:   Press ← with text in the answer box The bar appears. Click the bar The answer box is blank.   Seems like the answer isn't properly carried between the Fragments.  Either way, you lost everything you wrote.  Can somebody test this? And especially without a confirm dialog, it is a brush of the finger to lose a long post of code or something.   I hope this report will help because I like the app (I'd say 4/5 stars) but the bugs are preventing a solid 5 stars.   Stack Exchange 1.0.73 Samsung Galaxy S II Skyrocket I727 Android 4.1.2 I run a job every minute to reindex my site's content.   Today, the search engine died, and when I logged in there were hundreds of orphan processes that had been started by cron.  Is there another way using some kind of existing software that will let me execute a job every minute, but that won't launch another instance if that job doesn't return (i.e. because the search engine process has failed)? Show a message if directly accessing a deleted answer or comment Currently, if you access a deleted question you will get a short message informing you that the question has been deleted. However, if you click on a direct link to a deleted answer or comment (which you can get via notifications), nothing is displayed.  This is naturally quite confusing as you will get the message about a new answer or comment via notifications, click on the link and simply see nothing.  Therefore, I propose to display some sort of message to the user, informing her/him that the answer or question has been deleted. Note that this is different from always showing a deleted answer/comment, as I specifically request it in case you access the specific URL.  One example for a comment would be:   One example for a deleted answer is:   This question is related to ,  but requests a feature to circumvent the problem altogether. Infinitely many primes of the type 5 mod 6. Problem: Prove that there are infinitely many primes of the type 5 mod 6.  My professor did the problem and the proof was horribly long. Can someone show me a shorter version of the proof of this problem.