Calculate $ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n+1} H_n$ Define $$ H_n = \sum_{k=1}^n \frac{1}{k} $$ I need to calculate the sum $$ S = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n+1} H_n $$ Using the following integral representation of $ H_n$ $$ H_n = -n \int_0^1 x^{n-1}\ln(1-x) dx $$ and exchanging the order of summation, I obtained $$ S = -\int_0^1 \left(\frac{1}{1+x}+\frac{\ln(1+x)-x}{x^2}\right) \ln(1-x) dx $$ Using Wolfram Alpha, I got $$ S \approx 0.240227 $$ so I guess $$ S = \frac{(\ln 2)^2}{2} $$ But I don't know how to calculate the integral. Any idea?
On a series involving harmonic numbers Finding $$\sum^{\infty}_{n=1}\bigg[\bigg(\frac{(-1)^{n+1}}{n+1}\bigg)\bigg(\sum^{n}_{r=1}\frac{1}{r}\bigg)\bigg]$$   Try: Let $\displaystyle H_{n} =\sum^{n}_{r=1}\frac{1}{r},$ then series $\displaystyle \sum^{\infty}_{n=1}\frac{(-1)^{n+1}}{n+1}\cdot H_{n}=\frac{H_{1}}{2}-\frac{H_{2}}{3}+\frac{H_{3}}{4}\cdots\cdots$  could some help me to solve it, thanks
I want to argue that argue that $\pi(\alpha)=\alpha^3+3\alpha+3$ is an irreducible polynomial over the finite field with 5 elements $\mathbb{F}_5$. My approach was just to check that $\pi$ has no roots in $\mathbb{F}_5$. Is this right? If $\dfrac{4x^2-1}{4x^2-y^2}$ is an integer, then it is $1$ The problem is the following:     If $x$ and $y$ are integers such that $\dfrac{4x^2-1}{4x^2-y^2}=k$ is   also an integer, does it implies that $k=1$?   This equation is equivalent to $ky^2+(1-k)4x^2=1$ or to $(k-1)4x^2-ky^2=-1$. The first equation is a pell equation (if $k$ is a perfect square) and the second is a pell type equation (if $k-1$ is a perfect square). I've tried setting several values of $k$ to get some solutions but i got nothing. I'm starting to think that $k$ must be $1$. Indent math blocks? I want all math blocks \begin{math} in my document to be intended automatically, like it's the case with \begin{itemize}.   Can anyone please give me a hint on how to do this? Thanks :-) How can I get the DateTime for the start of the week? $ d = \gcd(a,b)\Rightarrow\ \gcd(a/d,b/d) = 1$ Could someone please help me with this proof?     Suppose that $a, b \in N$, and $d = \gcd(a, b)$. Since $d$ divides $a$, we have $a = de$ for some integer $e,$ and similarly $b = df$ for some integer $f$. Prove that $\gcd(e, f) = 1$.   I understand why it works. Since d is all the common factors of $a$ and $b, e$ and $f$ had no common factors, therefore the $\gcd(e,f) = 1$. But how do I prove this?  Thanks in advance. This was a question from my textbook. This is how I approached it.   For the first part, you can do c(10,5) so you choose 5 of the 10 to be on a team.   For the second part, you have 10 ways to choose the "best player" and you multiply it by c(10,4) to pick the other 4 players. However, I am not sure how you can exclude the "worst player". Although this sounds subjective, I believe the question is saying that the "best" and "worst" players are distinct, whereas the other players aren't. How to enumerate an enum How can you enumerate an enum in C#?  E.g. the following code does not compile:  public enum Suit {     Spades,     Hearts,     Clubs,     Diamonds }  public void EnumerateAllSuitsDemoMethod() {     foreach (Suit suit in Suit)     {         DoSomething(suit);     } }   And it gives the following compile-time error:     'Suit' is a 'type' but is used like a 'variable'   It fails on the Suit keyword, the second one. Using a java library from python I have a python app and java app. The python app generates input for the java app and invokes it on the command line.  I'm sure there must be a more elegant solution to this; just like using JNI to invoke C code from Java.  Any pointers? (FYI I'm v. new to Python)    Clarification (at the cost of a long question: apologies) The py app (which I don't own) takes user input in the form of a number of configuration files. It then interprits these and farms work off to a number of (hidden) tools via a plugin mechanism. I'm looking to add support for the functionality provided by the legacy Java app.  So it doesn't make sense to call the python app from the java app and I can't run the py app in a jython environment (on the JVM).  Since there is no obvious mechanism for this I think the simple CL invocation is the best solution. How do I mount shared folders in Ubuntu using VMware tools? Experts. I've successfully installed VMware tools for Ubuntu. Everything seems to work fine, but shared folders were not mounted automatically. How do I get them to work?  If I run vmware-hgfsclient in terminal, I get the list of shared folders, but ls -l /mnt/hgfs is empty. Actually there's no hgfs dir in /mnt. I know I should probably use the vmware-hgfsclient tool, but I realy don't know how.   P.S. I wouldn't ask if I could understand the vmware-hgfsclient help I've read.   Any suggestions? If $\sum_{n=1}^\infty \frac{1}{a_n}$ converges, must $\sum_{n=1}^\infty \frac{n}{a_1 + \dots + a_n}$ converge? Suppose $\sum_{n=1}^\infty \frac{1}{a_n} = A$ is summable, with $a_n &gt; 0,$ $n = 1,2,3,\cdots.$ How can we prove that $\sum_{n=1}^\infty \frac{n}{a_1 + \dots + a_n}$ is also summable?  This question came from a problem-solving seminar, but I'm quite stuck without a push in the right direction. I tried a few things, including Cauchy-Schwarz (which says $\sum_{n=1}^\infty \frac{n}{a_1 + \dots + a_n} &lt; \sum_{n=1}^\infty \frac{A}{n}$) and also the idea of assuming the latter series diverges and attempting to deduce the divergence of the former series from that, using facts such as $\sum a_n = \infty \implies \sum \frac{a_n}{a_1 + \cdots + a_n} = \infty$. Nothing has worked so far. Beamer and description environment text size Is there a single word for 'without a beginning'? From this  by orlp:     If I strap a rocket booster to a rocket in space and fire it for one second, then the work provided is much higher when the rocket is flying fast compared to when the rocket was stationary. In both cases the rocket fires for the same duration but in the former case the rocket travels a much greater distance during this period. What gives? How does one manually restart graphics drivers? I'm suffering from the dreaded . The problem is intermittent and can be "solved" by putting the computer to sleep or restarting it. I hope simply restarting the graphics drivers will remove the symptoms of the problem.  would seem to indicate that restarting the graphics drivers is possible in Windows 7. My question is: how? How can I re install the APT Tool? I was surprised to find today that I couldn't track down any simple way to write the contents of an InputStream to an OutputStream in Java. Obviously, the byte buffer code isn't difficult to write, but I suspect I'm just missing something which would make my life easier (and the code clearer).  So, given an InputStream in and an OutputStream out, is there a simpler way to write the following?  byte[] buffer = new byte[1024]; int len = in.read(buffer); while (len != -1) {     out.write(buffer, 0, len);     len = in.read(buffer); } Accessing System Databases/Tables using LINQ to SQL? CSS - Make divs align horizontally I have a container div with a fixed width and height, with overflow: hidden.  I want a horizontal row of float: left divs within this container. Divs which are floated left will naturally push onto the 'line' below after they read the right bound of their parent. This will happen even if the height of the parent should not allow this. This is how this looks:  ![Wrong][1] - removed image shack image that had been replaced by an advert  How I would like it to look:  ![Right][2] - removed image shack image that had been replaced by an advert  Note: the effect I want can be achieved by using inline elements &amp; white-space: no-wrap (that is how I did it in the image shown). This, however, is no good to me (for reasons too lengthy to explain here), as the child divs need to be floated block level elements. Setting Boundaries around city in google maps Minimum value of $\left(a+\frac{1}{b}\right)^2+\left(b+\frac{1}{c}\right)^2+\left(c+\frac{1}{a}\right)^2$ Given $a,b,c \in \mathbb{R^+}$ such that $a+b+c=12$   Find Minimum value of $$S=\left(a+\frac{1}{b}\right)^2+\left(b+\frac{1}{c}\right)^2+\left(c+\frac{1}{a}\right)^2$$  My Try: By Cauchy Schwarz Inequality we have  $$\left(a+\frac{1}{b}\right)+\left(b+\frac{1}{c}\right)+\left(c+\frac{1}{a}\right)\le \sqrt{3}\sqrt{S}$$  $\implies$  $$\sqrt{3S} \ge 12+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$$  Now by $AM \ge HM $ inequality we have  $$\frac{a+b+c}{3} \ge \frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}$$ $\implies$  $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \ge \frac{3}{4}$$  hence  $$\sqrt{3S} \ge 12+\frac{3}{4}=\frac{51}{4}$$  hence  $$3S \ge \frac{2601}{16}$$  so $$S \ge \frac{867}{16}$$  is this approach correct and any better approach please share.