Is $\sum\limits_{n = 1}^\infty (-1)^n\frac{H_n}{n}=\frac{6\ln^2(2)-\pi^2}{12}$ a valid identity? A while ago I asked a question about an identity that I found while playing with series involving harmonic numbers. However, since the methods I used were not the focus of my question, I never explained how I got this result. So here it goes...  NOTE: It may be of help to work backwards through this problem instead. This is just how I did it.  We start by defining $H_n$ as the $n^{th}$ harmonic number. We also take note of the identities:  $$ \sum_{n=1}^\infty \frac{(-1)^n}{n} = -\ln(2)$$ $$ \sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$$  Now we note that:  $$ \tag{1}H_n = \sum_{k=1}^\infty \frac{1}{k} - \frac{1}{k+n} = \sum_{k=1}^\infty \frac{n}{k(k+n)}$$  Since $\sum_{k=1}^\infty \frac{1}{k} - \frac{1}{k+n}$ is a telescoping series.  Dividing by $n$ we get: $$\tag{2} \frac{H_n}{n} = \sum_{k=1}^\infty \frac{1}{k(k+n)}$$  Since $ \sum_{n=1}^\infty\frac{H_n}{n}$ is obviously divergent, I took a look at the alternating series:  $$ \tag{3}\sum_{n=1}^\infty(-1)^n\frac{H_n}{n} = \sum_{n=1}^\infty(-1)^n\left(\sum_{k=1}^\infty \frac{1}{k(k+n)}\right)$$  We must now multiply this series by 2 and add $\sum_{n=1}^\infty\frac{1}{n^2} $ which is necessary for the next step.  So we have:  $$\begin{align}\tag{4}\sum_{n=1}^\infty\frac{1}{n^2} + 2 \sum_{n=1}^\infty(-1)^n\left(\sum_{k=1}^\infty \frac{1}{k(k+n)}\right) &amp; = \left(\frac{1}{1}\cdot\frac{1}{1} + \frac{1}{2}\cdot\frac{1}{2} + \frac{1}{3}\cdot\frac{1}{3}+\cdots\right) \\ &amp; -2 \left(\frac{1}{1}\cdot\frac{1}{2} + \frac{1}{2}\cdot\frac{1}{3} + \frac{1}{3}\cdot\frac{1}{4}+\cdots\right)  \\ &amp; +2 \left(\frac{1}{1}\cdot\frac{1}{3} + \frac{1}{2}\cdot\frac{1}{4} + \frac{1}{3}\cdot\frac{1}{5}+\cdots\right)  \\ &amp; -2 \left(\frac{1}{1}\cdot\frac{1}{4} + \frac{1}{2}\cdot\frac{1}{5} + \frac{1}{3}\cdot\frac{1}{6}+\cdots\right) \\ &amp; + \cdots \end{align}$$  With a very careful (and possibly illegal) rearrangement of this expanded sum, we can finally obtain: $\\$ (NOTE: It may be easier to work backwards through this step)  $$ \begin{align} \tag{5}\sum_{n=1}^\infty\frac{1}{n^2} + 2 \sum_{n=1}^\infty(-1)^n\left(\sum_{k=1}^\infty \frac{1}{k(k+n)}\right) &amp;= \left(-\frac{1}{1} + \frac{1}{2} - \frac{1}{3} + \cdots\right)\left(-\frac{1}{1} + \frac{1}{2} - \frac{1}{3} + \cdots\right) \\ &amp; = \ln^2(2)\end{align}$$  Thus, we have:   $$\tag{6}\sum_{n=1}^\infty(-1)^n\frac{H_n}{n} = \frac{6\ln^2(2) - \pi^2}{12} = -0.5822...$$   are 3 graphs from computing the actual value of the sum. The first graph is of the difference between the actual sum and  $\frac{6*ln^2(2)-\pi^2}{12}$ for k up to 14000. As you can see it seems to converge on 0. You can ignore the 2nd and 3rd graphs.  QUESTIONS:  Is the method used from step 4-5 (or vise versa) legal even though it involves conditionally convergent series? If not, is there any restrictions and/or rearrangements I can use to make it legal?  Is there any known generalizations for $\sum_{n=1}^\infty(-1)^n\frac{H_n}{n^q}$ that extend this result?  Recommended readings that cover this topic would be greatly appreciated as well.  Thanks, Dom
Proving an alternating Euler sum: $\sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2$ Let $$A(p,q) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}H^{(p)}_k}{k^q},$$ where $H^{(p)}_n = \sum_{i=1}^n i^{-p}$, the $n$th $p$-harmonic number.  The $A(p,q)$'s are known as alternating .     Can someone provide a nice proof that    $$A(1,1) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2?$$   I worked for a while on this today but was unsuccessful.  Summation by parts, swapping the order of summation, and approximating $H_k$ by $\log k$ were my best ideas, but I could not get any of them to work.  (Perhaps someone else can?)  I would like a nice proof in order to complete .  Bonus points for proving $A(1,2) = \frac{5}{8} \zeta(3)$ and $A(2,1) =  \zeta(3) - \frac{1}{2}\zeta(2) \log 2$, as those are the other two alternating Euler sums needed to complete my answer.   Added: I'm going to change the accepted answer to robjohn's $A(1,1)$ calculation as a proxy for the three answers he gave here.  Notwithstanding the other great answers (especially the currently most-upvoted one, the one I first accepted), robjohn's approach is the one I was originally trying.  I am pleased to see that it can be used to do the $A(1,1)$, $A(1,2)$, and $A(2,1)$ derivations.
Solve a congruence $1978^{20}\equiv x\pmod{125}$ Is there a problem with defrosting meat on the counter? I generally defrost meat on the kitchen counter. A friend of mine suggested that this was dangerous and suggested that I defrost meat in the refrigerator. I am no biologist but it seems to me that as long as the meat doesn't get warm defrosting it on the counter should be safe.  Generally, I remove the frozen items from the freezer and place them on a plate on the counter in the packaging they were frozen in. When they are mostly / completely thawed I place the meat in the refrigerator.  Am I wrong and should I be defrosting in the refrigerator? Should 'Hi', 'thanks', taglines, and salutations be removed from posts? I edit a lot of posts every day. I often run across posts with 'Hi' and 'Thanks' on the top and the bottom of the post respectively. I also run across things like:     --User   Should these items be removed during the editing of the post by an editor? Where can I find computer modern fonts for use in other programs, e.g. Adobe Illustrator? For some complicated diagrams, I need to use something like Adobe Illustrator or Inkscape, and would like to include nicely typeset labels. (I'm aware of post-processing solutions, such as pinlabel.) One way to do this is to use a closely-matching font, and then convert fonts to outlines. Better, however, would be to use Computer Modern directly in Illustrator.  Where can I find a Computer Modern font compatible with Adobe Illustrator? Why every map $f : S^n \to T^n (n>1)$ has topological degree zero? Why every map $f : S^n \to T^n (n&gt;1)$ has topological degree zero?  I don't know anything about covering spaces, and has been told to me that this assertion comes from this theory!  I do appreciate any help. $\newcommand{\Var}{\operatorname{Var}}$In this  is claimed that if the equation of errors in OLS is given by: $$u=y - X\beta$$ Then in the presence of heteroscedasticity the variance of $u$, will not be constant, $\sigma^2 \times I$, where $I$ is an identity matrix, but: $$\Var(u\mid X)=\sigma^2\Omega$$ In order to account for the heteroskedasticity, we can estimate the transform system, such that $P$ is a transformation matrix. $$Py=PX\beta-Pu$$ Where "the variance of a constant matrix $P$ times a random vector $u$" is: $$\Var(Pu\mid X)=P\Var(u\mid X)P'=P(\sigma^2\Omega)P'$$ Can somebody explain me the proof for that? Prove that $|f''(\xi)|\geqslant\frac{4|f(a)-f(b)|}{(b-a)^2}$ Download all PDF links in a web page? Do you know a good software to download all PDF links in a web page??  Operating system is Windows 7. Is it possible to show our custom message in QGIS Status bar using python? Difference between phrase and idiom What is the difference between a phrase and an idiom? What's the blue pentagon for in Pokemon XY? On the Pokemon status screen, underneath the Pokemon and to the left of the custom markings, is a small blue pentagon:  This doesn't seem to be present on Pokemon obtained through the Pokemon Transporter from Gen V:  I've looked into it myself, and popular opinion seems to be that it's either some kind of &quot;Legitimate Pokemon&quot; mark or simply to indicate that the Pokemon originates from Gen VI. The latter seems far more likely to me, but do we know for sure what this blue hexagon means? More homework help.  Given the function $f:A \to B$.  Let $C$ be a subset of $A$ and let $D$ be a subset of $B$.  Prove that:  $C$ is a subset of $f^{-1}[f(C)]$  So I have to show that every element of $C$ is in the set $f^{-1}[f(C)]$  I know that $f(C)$ is the image of $C$ in $B$ and that $f^{-1}[f(C)]$ is the pre-image of $f(C)$ into $A$.  Where I'm stuck is how to use all of this information to show/prove that $C$ is indeed a subset.    Do I start with an arbitrary element (hey, let's call it $x$) of $C$? and then show that $f^{-1}[f(x)]$ is $x$?  I could use a little direction here...  Thanks. How to pin either a Shortcut or a Batch file to the new Windows 7, 8 and 10 Taskbar and start menu? We are having trouble adding our batch scripts to the Windows 7, 8 or 10 taskbar or start menu.  Our batchfiles take some arguments and just execute other application based on those arguments . To keep it simple, we created some shortcuts for our customers which they just need to click in order to get everything running.  The question here is: How can we pin those shortcuts or batch files to the taskbar or start menu?  (The customers aren't supposed to access the file system and that's why they can only see the taskbar) Photons have no mass but they can push things, as evidenced by .   Can photons push the source which is emitting them? If yes, will a more intense flashlight accelerate me more? Does the wavelength of the light matter? Is this practical for space propulsion? Doesn't it defy the law of momentum conservation?  Note: As John Rennie mentioned, all in all the wavelength doesn't matter, but for a more accurate answer regarding that, see the comments in  .  Related Wikipedia articles: , How can I find maximum power transfer using Thevenin equivalent Useless math that became useful I'm writing an article on  and some people pointed out that this is completely useless.  My idea is to amend my article with some theories that seemed useless when they are created but found use after some time.  I came with some ideas like the Turing machine but I think I'm not grasping the right examples.  Can someone point me some theories that seemed like the Lychrel numbers and then become 'useful'? In statistical learning, implicitly or explicitly, one always assumes that the training set $\mathcal{D} = \{ \bf {X}, \bf{y} \}$ is composed of $N$ input/response tuples $({\bf{X}}_i,y_i)$ that are independently drawn from the same joint distribution $\mathbb{P}({\bf{X}},y)$ with  $$ p({\bf{X}},y) = p( y \vert {\bf{X}}) p({\bf{X}}) $$  and $p( y \vert {\bf{X}})$ the relationship we are trying to capture through a particular learning algorithm. Mathematically, this i.i.d. assumption writes:  \begin{gather} ({\bf{X}}_i,y_i) \sim \mathbb{P}({\bf{X}},y), \forall i=1,...,N \\ ({\bf{X}}_i,y_i) \text{ independent of } ({\bf{X}}_j,y_j), \forall i \ne j \in \{1,...,N\} \end{gather}  I think we can all agree that this assumption is rarely satisfied in practice, see this related  and the wise comments of @Glen_b and @Luca.  My question is therefore:      Where exactly does the i.i.d. assumption becomes critical in practice?    [Context]   I'm asking this because I can think of many situations where such a stringent assumption is not needed to train a certain model (e.g. linear regression methods), or at least one can work around the i.i.d. assumption and obtain robust results. Actually the results will usually stay the same, it is rather the inferences that one can draw that will change (e.g. heteroskedasticity and autocorrelation consistent HAC estimators in linear regression: the idea is to re-use the good old OLS regression weights but to adapt the finite-sample behaviour of the OLS estimator to account for the violation of the Gauss-Markov assumptions).   My guess is therefore that the i.i.d. assumption is required not to be able to train a particular learning algorithm, but rather to guarantee that techniques such as cross-validation can indeed be used to infer a reliable measure of the model's capability of generalising well, which is the only thing we are interested in at the end of the day in statistical learning because it shows that we can indeed learn from the data. Intuitively, I can indeed understand that using cross-validation on dependent data could be optimistically biased (as illustrated/explained in ).   For me i.i.d. has thus nothing to do with training a particular model but everything to do with that model's generalisability. This seems to agree with a paper I found by Huan Xu et al, see "Robustness and Generalizability for Markovian Samples" .   Would you agree with that?  [Example]   If this can help the discussion, consider the problem of using the LASSO algorithm to perform a smart selection amongst $P$ features given $N$ training samples $({\bf{X}}_i,y_i)$ with $\forall i=1,...,N$ $$ {\bf{X}}_i=[X_{i1},...,X_{iP}] $$ We can further assume that:   The inputs ${\bf{X}}_i$ are dependent hence leading to a violation of the i.i.d. assumption (e.g. for each feature $j=1,..,P$ we observe a $N$ point time series, hence introducing temporal auto-correlation) The conditional responses $y_i \vert {\bf{X}}_i$ are independent. We have $P \gg N$.    In what way(s) does the violation of the i.i.d. assumption can pose problem in that case assuming we plan to determine the LASSO penalisation coefficient $\lambda$ using a cross-validation approach (on the full data set) + use a nested cross-validation to get a feel for the generalisation error of this learning strategy (we can leave the discussion concerning the inherent pros/cons of the LASSO aside, except if it is useful). Code formatting - tag/block rather than indentation - code block more obvious I'm calculating the covariance of a distribution in parallel and I need to combine the distributed results into on singular Gaussian. How do I combine the two?  Linearly interpolating between the two almost works, if they are similarly distributed and sized.   provides a forumla at the bottom for combination but it doesn't seem right; two identically distributed distributions should have the same covariance, but the formula at the bottom of the page doubles the covariance.  Is there a way to combine two matrices? Labelled in-line equation