Finding the infinite Sum of a series: $\sum\frac1{n(n+1)(n+2)}$ Find the infinite Sum of the series with general term $\frac{1}{n(n+1)(n+2)}$.  I decomposed the fraction upto this $1/(2n)-1/(n+1)+1/(2n+4)$. But I find no link about cancelling terms. So how should I find the infinite Sum?   Please don't say that this is off topic or homework.  if wanna say that this is homework, at least give a hint if I am going the right way?
Find the sum of the series $\sum \frac{1}{n(n+1)(n+2)}$ I got this question in my maths paper     Test the condition for convergence of $$\sum_{n=1}^\infty \frac{1}{n(n+1)(n+2)}$$   and find the sum if it exists.   I managed to show that the series converges but I was unable to find the sum. Any help/hint will go a long way. Thank you.
Show $\operatorname{Aut}(C_2 \times C_2)$ is isomorphic to $S_3=D_6$ Script Tag Error - Aura Lightning Component Is it possible to recover an email that has been clicked "delete forever" in the trash folder? Does $\sum \limits_{n=1}^\infty\frac{\sin n}{n}(1+\frac{1}{2}+\cdots+\frac{1}{n})$ converge (absolutely)? I've had no luck with this one. None of the convergence tests pop into mind.  I tried looking at it in this form $\sum \sin n\frac{1+\frac{1}{2}+\cdots+\frac{1}{n}}{n}$ and apply Dirichlets test. I know that $\frac{1+\frac{1}{2}+\cdots+\frac{1}{n}}{n} \to 0$ but not sure if it's decreasing.  Regarding absolute convergence, I tried:  $$|\sin n\frac{1+\frac{1}{2}+\cdots+\frac{1}{n}}{n}|\geq \sin^2 n\frac{1+\frac{1}{2}+\cdots+\frac{1}{n}}{n}=$$  $$=\frac{1}{2}\frac{1+\frac{1}{2}+\cdots+\frac{1}{n}}{n}-\frac{1}{2}\cos 2n\frac{1+\frac{1}{2}+\cdots+\frac{1}{n}}{n}$$  But again I'm stuck with $\cos 2n\frac{1+\frac{1}{2}+\cdots+\frac{1}{n}}{n}$. Assuming it converges then I've shown that $\sum \sin n\frac{1+\frac{1}{2}+\cdots+\frac{1}{n}}{n}$ doesn't converge absolutely. When and why is the [featured] tag removed by the Community user? I've found that the Meta (moderator only) tag  is automatically  by the  user. So, I want to know when and why is the tag  is removed by this user?  Is there any mechanism for this removal or is there simply a specified time period after which the  is automatically removed by the Community user? No error message if the URL is very long I entered a URL with a very long parameter on Stack Exchange.  I have type this url    It shows a blank screen , does not shown any messages.    We need to show a “Page Not Found” message, same as what you did on other Stack Exchange pages. I need a step-by-step, simple and easy way to configure swappiness. Misplaced navigation bar and question title It appears that on Server Fault and Super User, the navigation bar and question title have suddenly become misplaced. The same does not occur on SO and mSO, nor on any of the other SE sites I spot-checked.  This is being seen in both Chrome and Firefox and just began occurring within the last few minutes.      What it should look like:      While this was fixed for a few hours, it has returned as of 22:05. Area of intersection between two circles Suppose you have 2 circles that intersect each other in such a way that each circle passes through the other's center. What is the area between the circle(or common area) i.e. area between the centres of the circles? To whom was Yoda referring when he said, "There is another?" In the original trilogy Yoda refers to another hope:     Obi-Wan: "That boy is our last hope."   Yoda: "No...there is another..."   The assumption being after seeing Return of the Jedi (Episode VI) that the other was Leia.  If, however, you consider the prequel trilogy canon, then this assumption depends greatly on your point of view*. Obi Wan knows about Leia (and for unknown reasons has discounted her as a hope), so Yoda must be referring to someone or something else.  Is there any canonical source as to what Yoda meant, or was uncle George setting us up for Episode VII?  *as a wise man once said      you will find that many of the truths we cling to depend greatly on   our own point of view.   Just to add a bit of clarity: I was thinking about who Yoda was referring to when he was talking to Obi-Wan. When talking to Luke, it is obvious that Leia is the other (as he talks about a 'Skywalker'). It is not so obvious (after watching the prequel trilogy) that he means it when talking to Obi-Wan. When posting an HTTPS url like https://stackoverflow.com/..., it does not onebox in chat. For example: OneDrive isn't uploading all my photos from my phone I've got an Office 365 (Microsoft 365) Family license. I've set up OneDrive on my Samsung Galaxy S10 phone. I've verified that it is set up by following the . I've got it set up to upload photos using either Wi-Fi or mobile. And I see photos I've taken with my phone, uploaded to OneDrive from as recently as a few days ago. So, it works. But not all the time or for all my photos. I've scrolled through my photos on my phone's Gallery app. I see some photos from as recently as 4 days ago to several months ago, are not uploaded to OneDrive. Why is that? If $G$ is abelian, then the set of all $g \in G$ such that $g = g^{-1}$ is a subgroup of $G$ Prove that if $G$ is abelian then the set $H$ of all elements of $G$ that are their own inverses is a subgroup of $G$.   Naturally in an abelian group, $ab = ba$ for $a, b \in G$, however I'm not sure how to show the set elements that are their own inverses is a subgroup of $G$ using arbitrary elements. I came across the following question:    This can be easily flagged for too broad or opinion based, but I wonder on which stack-exchange site can one ask answers for such a question? Troubleshoot High CPU usage by the "System" process I have noticed that from some time my system is freezing and its probably caused by the high CPU usage which is caused by the system process.  All applications I'm running is the Skype, TeamSpeak and Chrome so it definitely shouldnt eat that amount of CPU.  You can see the problem itself and running processes in the screenshot below:    Sometimes CPU usage is reaching 90%, but the average usage is like 40-65%.  My PC parameters:   Windows 8 (customer preview) Intel Core i3 - 2350M 8 GB RAM   I'd appreciate any help attempt! Regards.  --UPDATE--  As the user below posted a great answer, I have noticed that the process that is eating the most CPU in the system is called Arthurx.sys, simple google tells that it's a TPLink driver (an wifi adapter, I have bought like 2 weeks ago!) drivers has been installed from the Windows MSDN, but also tried to install the drivers from the attached CD, but it doesn't help. From the system start, it is using like 5% of the CPU only, but after 2-4 hours of working it is growing up and reaching 40-60% of the CPU usage.  Device name: TPLink WN722N I generally know how to use a hyphen, but when should I use an en-dash (–) instead of an em-dash, or when should I use a hyphen (-) instead of an em-dash (—)? How can top principal components retain the predictive power on a dependent variable (or even lead to better predictions)? Suppose I am running a regression $Y \sim X$. Why by selecting top $k$ principle components of $X$, does the model retain its predictive power on $Y$?   I understand that from dimensionality-reduction/feature-selection point of view, if $v_1, v_2, ... v_k$ are the eigenvectors of covariance matrix of $X$ with top $k$ eigenvalues, then $Xv_1, Xv_2 ... Xv_k$ are top $k$ principal components with maximum variances. We can thereby reduce the number of features to $k$ and retain most of the predictive power, as I understand it.   But why do top $k$ components retain the predictive power on $Y$?   If we talk about a general OLS $Y \sim Z$, there is no reason to suggest that if feature $Z_i$ has maximum variance, then $Z_i$ has the most predictive power on $Y$.   Update after seeing comments: I guess I have seen tons of examples of using PCA for dimensionality reduction. I have been assuming that means the dimensions we are left with have the most predictive power. Otherwise what's the point of dimensionality reduction? velocity of an object after leaving a ramp Punctuation with units I remember reading somewhere that if a unit is abbreviated as one character, there must not be a space between the number and the unit (e.g., 5m, 26K). If the unit is abbreviated as two or more Characters, there must be a space between the number and the unit (e.g., "10 km", "USD 5").   Can you please help me find the source again? Is this recommendation correct? Immutability of Strings in Java