Show that $\displaystyle\sum^{\infty}_{n=1}\frac{(-1)^n}{n}$ converges.    Hint: consider the sum of two subsequent entries.   How can I use that hint to solve the questaion.. ?
I know that the harmonic series $$\sum_{k=1}^{\infty}\frac{1}{k}  = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \cdots + \frac{1}{n} + \cdots \tag{I}$$ diverges, but what about the alternating harmonic series   $$\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}  = \frac{1}{1} - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \cdots + \frac{(-1)^{n+1}}{n} + \cdots \text{?} \tag{II}$$  Does it converge?   If so, what is its sum?
Same gravatar is displayed for different users in the stackexchange.com , under the Top Network Askers in the right side bar.   The red circled image's source (&lt;img src=) are from   https://www.gravatar.com/avatar/?s=18&amp;d=identicon&amp;r=PG&amp;s=30                                ^   Seems the gravatar id is missing here in the image source.  For an example from the list, the user  gravatar id should be    https://www.gravatar.com/avatar/66c66c4d97a7bc3859a85253f003e1fb?s=18&amp;d=identicon&amp;r=PG&amp;s=30   but it is mapped with   https://www.gravatar.com/avatar/?s=18&amp;d=identicon&amp;r=PG&amp;s=30   which caused the display issue. This is a small exercise that I just can't seem to figure out. When I see it I'll probably go 'ahhh!', but so far I haven't made any progress.  I'd like to prove that any linear functional $\phi$ on $\ell^\infty$ that is positive, i.e. for any $(x_n) \in \ell^\infty$ such that $x_n \geq 0$ for all $n \in \mathbb{N}$ we have $\phi((x_n)) \geq 0$, is a bounded linear operator and has $\|\phi \| = \phi((1,1,...))$.  Nothing I have tried worked so far, so any hints would be appreciated, thanks! Extension of degree 2 with characteristic not 2 I am told that $K \leq L$ is a field extension of degree 2 and that $K$ has characteristic not 2. I am then asked to show that $\exists \: \alpha \in L$ such that $L = K(\alpha), \; \alpha^2\in K $  So far I have that:  $\exists \; \alpha_1, \alpha_2 \in L$ such that $L = K( \alpha_1, \alpha_2)$  $\Rightarrow |L:K| = |L:K(\alpha_1)| |K(\alpha):K| $ so $|K(\alpha):K| = 1$ or $2$  $|K(\alpha_1):K| = 2 \Rightarrow L = K(\alpha_1)$ and $deg(f_{\alpha_1}(t)) = 2  $ where $f_{\alpha_1}(t)$ is the minimal polynomial of $\alpha_1$ over $K$   So $f_{\alpha_1}(t) = t^2 + at + b$ for some $a,b \in K$  Also, $|K(\alpha_1):K|=1 \Rightarrow K(\alpha_1) = K$ and $L = K(\alpha_2)$ which means $degf_{\alpha_2}(t) = 2$ where $f_{\alpha_2}(t)$ is the minimal polynomial of $\alpha_2$ over $K$.   So $f_{\alpha_2}(t) = t^2 + at + b$ for some $a,b \in K$  Given that we consider the same polynomial each time, relabel to consider:   $f_{\alpha}(t) = t^2 + at+ b $ for some $a,b \in K$  We have already demonstrated that $L=K(\alpha)$ for some $\alpha \in L$ so now we much show that $\alpha^2 \in K$  $\alpha^2 \in K \Leftrightarrow a = 0_K$  I now have no idea how to show that this coefficient of $f_{\alpha}(t)$ is necessarily $0$.   I know that $f_{\alpha}(t)$ has two roots in $L$ and that because $K$ and thus $L$ has characteristic not 2, they cannot be the same root, but I don't know how to then conclude that they must sum to $0$  Any help would be greatly appreciated. When should I use "didn't" instead of "haven't"? Is haven't supposed to mean something I may still do and didn't not?  Can I use didn't for something I might still do? is a simple way to script small GUIs.  Can anyone give a nice example with a button and a text widget. When the button is pressed should a shell command be executed and the output piped to the text widget.  If you have other nice and clean examples for useful tasks, please add them too. When an option expires does that constitute a buy or sell A similar question recently asked about stock sales and option buys triggering wash sale rules.  If an option expires, is that considered a buy/sale ?  What I mean is: let's assume I have a call option with a target date of August 16. Now I sell the underlying stock somewhere before that (after July 16th). Does that trigger wash sale rules (if the option expires in the money, exercised by default) ?  Is an option expiring out of the money considered a sale ? Is it possible to use the startup.blend UI layout in older files? Does a non-trivial solution exist for $f'(x)=f(f(x))$? How can I enable multiple users login at a time in remote system? Recently we bought dedicated server with Windows Server 2012 Standard.It's allowing 2 people at a time.So I want to allow more people at a time.  How can I do this? I was working on a Beamer presentation, and noticed that, if I open a group scope and use a \tiny, the font size will bet set, but there will be a large space between lines. However, if I issue a \par at the end of the paragraph, it works as intended:  \documentclass[t]{beamer} \newcommand{\shortlipsum} {Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Ut purus elit, vestibulum ut, placerat ac, adipiscing vitae, felis. Curabitur dictum gravida mauris. Nam arcu libero, nonummy eget, consectetuer id, vulputate a, magna. Donec vehicula augue euneque. Pellentesque habitant morbi tristique senectus et netus et malesuada fames ac turpis egestas. Mauris ut leo. Cras viverra}  \begin{document} \begin{frame} \shortlipsum  {\tiny \shortlipsum\par   % &lt;--- removing this \par will break line spacing } \end{frame} \end{document}   Here's a comparison between not using and using \par:    Am I doing it the right way? Why is \par the right way? "Always on top" option missing for Firefox/Chrome in 18.04 How do you disable an application specific hotkey, for example, "Command-P"? In many applications, Command-P is used to print.  My fingers are accustomed to emacs settings and reflexively press Command-P not for printing but for browsing history. It's annoying to see the print pop-up window every time when I press Command-P. I don't need a shortcut to print. How do you block a shortcut globally? What's the best way of handling permissions for Apache 2's user www-data in /var/www? Has anyone got a nice solution for handling files in /var/www? We're running Name Based Virtual Hosts and the Apache 2 user is www-data.  We've got two regular users &amp; root. So when messing with files in /var/www, rather than having to...  chown -R www-data:www-data   ...all the time, what's a good way of handling this?  Supplementary question: How hardcore do you then go on permissions?  This one has always been a problem in collaborative development environments. Why do heavier objects fall faster in air? I'd like to run a .sh file using bash on logout and shutdown of Ubuntu. I have searched up and down the web and have found only fixes for KDE and GNOME. The script is simple, I just need to know if I can add the line of code to the "logoff/shutdown" file or if I need to reference it in another script. How do defenders of libertarian freewill reconcile it with constraints imposed by the laws of physics? It is fine to say that for an object flying past a massive object, the spacetime is curved by the massive object, and so the object flying past follows the curved path of the geodesic, so it "appears" to be experiencing gravitational acceleration. Do we also say along with it, that the object flying past in reality exeriences NO attraction force towards the massive object? Is it just following the spacetime geodesic curve while experiencing NO attractive force?  Now come to the other issue: Supposing two objects are at rest relative to each other, ie they are not following any spacetime geodesic. Then why will they experience gravitational attraction towards each other? E.g. why will an apple fall to earth? Why won't it sit there in its original position high above the earth? How does the curvature of spacetime cause it to experience an attraction force towards the earth, and why would we need to exert a force in reverse direction to prevent it from falling? How does the curvature of spacetime cause this?  When the apple was detatched from the branch of the tree, it was stationary, so it did not have to follow any geodesic curve. So we cannot just say that it fell to earth because its geodesic curve passed through the earth. Why did the spacetime curvature cause it to start moving in the first place? 'bridges' on a PCB copper layer I've come across a design where each pad was connected using 4 'bridges' to the GND copper layar. What stands behind these 'bridges'? Why not make a full copper layer with only solder-mask defining the pads? How can blackbody radition be explained by quantization? I don't understand why quantization makes a peak on the blackbody radiation curve (so there is no ) and the relationship between that peak and quantization concept.  When the blackbody is heated, it starts to glow. All atoms start to vibrate. Total heating energy must be divided and equally shared by all atoms of blackbody. Right?  But what happens next?  Do some atoms get much more energy to radiate at higher frequency and waste the energy? What happens at the atomic level, so quantization can explain it? What does classical physics claim, so it can not explain the radiation curve? Looking for a post-apocalyptic trilogy where pilot is nursed back to health by natives of the land I read a series of books around 2005. I think it might have been a trilogy.  Basic story line is in a post apocalyptic world. The main character is a pilot for the Federation or something similar to that. He crash lands, is nursed back to health by the natives, comes to like and respect them, and does not want to fight against them.  Also in the later book I think it also involves another race who are very Japanese-ish. They drink sake and the "clan" names are things like Fu-Jitsu and Mitsu-Bishi.