I know that these are some existing sums that are true: $$\sum_{r=1}^{n}r = \frac{n(n+1)}{2} = \mathcal{O}(n^2)$$ $$\sum_{r=1}^{n}r^2 = \frac{n(n+1)(2n+1)}{6} = \mathcal{O}(n^3)$$ $$\sum_{r=1}^{n}r^3 = \frac{n^2(n+1)^2}{4} = \mathcal{O}(n^4)$$  I may be butchering the definition of Big O' Notation, but I believe it describes the form of the function, i.e. the highest power of $n$ in these cases. Is there a proof for: $$\sum_{r=1}^{n}r^k = \mathcal{O}(n^{k+1})$$ since there seems to be this pattern?  Or even better, is there a general formulae that mirrors $$\sum_{r=1}^{n}r^k$$ in terms of only $n$ and $k$?  I didn't know how to phrase the question very well on Google, so I apologise if this question has been answered many times.
I have been wanting to understand how to find the sum of this series.   $$1^p + 2^p + 3^p +{\dots} + n^p$$  I am familiar with Gauss' diagonalised adding trick for the sum of the first $n$ natural numbers.  I can prove the formulas for  $$\begin{align}   \sum_{1}^{n} k^2 &amp;= \frac{n(n+1)(2n+1)}{6}\\  \sum_{1}^{n} k^3 &amp;= \frac{n^2(n+1)^2}{4}\\ \sum_{1}^{n} k^4 &amp;= \frac{n(n+1)(2n+1)(3n^2+3n-1)}{30} \\ \end{align}$$  With mathematical induction. But, beyond that even proofs with mathematical induction are difficult.  I'm interested in learning the theory and the proof behind Faulhaber's formula. What is the knowledge required to understand this proof ?
I have a wireless Access point which is connected to internet and the same being accessing by 2 system, one on windows XP and the other on Ubuntu.  Please let me know if it is possible to access the c:\ d:\ or any drives on windows xp machine directly from ubuntu system ?  I have tried doing this with 2 windows xp system but not sure how to do this from ubuntu.  Any help regarding the same is highly appreciated. (reason to do so is to access files in windows xp machine)  -Thanks &amp; regards, Manju To prevent overfitting people people add a regularization term (proportional to the squared sum of the parameters of the model) with a regularization parameter $\lambda$ to the cost function of linear regression. Is this parameter $\lambda$ the same as a lagrange multiplier? So is regularization the same as the method of lagrange multiplier? Or how are these methods connected? Should we `enable` the User Systemd service `pulseaudio.service`? Setting Objects to Null/Nothing after use in .NET I am looking for shapefiles or a list of suburb names for Melbourne and Canberra. I have ABS shapefiles, but I need to filter that to end up with only metropolitan suburbs.  My shapefile has codes in it for  - I just need to know which suburbs belong to the metros. Classical car collision I have a very confusing discussion with a friend of mine.  2 cars ($car_a$ and $car_b$) of the same mass $m$ are on a collision course. Both cars travel at $50_\frac{km}{h}$ towards each other.  They collide. Ignoring any shreds and collateral damage, what is the speed of collision that the driver of $car_a$ felt?  What I mean is, if $car_a$ were to be driven into an infinite mass wall, what would the velocity be to replicate the damage caused by the initial collision? How can we connect a PHP script to MS Access (.mdb) file?  I tried by including following PHP code:  $db_path = $_SERVER['DOCUMENT_ROOT'] . '\WebUpdate\\' . $file_name . '.mdb'; $cfg_dsn = "Provider=Microsoft.Jet.OLEDB.4.0;Data Source=" . $db_path; $odbcconnect = odbc_connect($cfg_dsn, '', '');   But it failed and I received following error message:   Warning: odbc_connect() [function.odbc-connect]: SQL error: [Microsoft][ODBC Driver Manager] Data source name not found and no default driver specified, SQL state IM002 in SQLConnect in C:\web\WebUpdate\index.php on line 41 Differences in development between .NET and Mono Given the following data:  972, 975, 985, 993, 993, 995, 998, 1001, 1004, 1007, 1008, 1009, 1011, 1015, 1016, 1020, 1022, 1032  I calculate the lower quartile as follows:  Number of values: 18, Number of 'gaps': 17.  Lower quartile is at 17/4 + 1 = 5.25th value. 5.25th value is 993 + (995 - 993) x 0.25 = 993.5  Upper quartile is at 3 x 17/4 + 1 = 13.75th value. 13.75th value is 1011 + (1015 - 1011) x 0.75 = 1014  Both the Excel QUARTILE function and the R quantile function agree with me. However the book I am using (Understanding Statistics by Graham Upton and Ian Cook) give different answers (993 and 1015 respectively). I'm confused. Which is correct? How to generate Bernoulli random variables with common correlation $\rho$? What function satisfies $F'(x) = F(2x)$? Alternative to LibreOffice Base for Ubuntu Why does the CAN bus use a 120 ohm resistor as the terminating resistor and not any other value? I know the reasons for using terminating resistors on a  and how important it is.  But why 120&nbsp;ohm? How did this value come up? Is there any specific reason to use 120&nbsp;ohm? I used WolframAlpha to calculate a sum but it didn't show me the way :( Anybody has a hint or a solution for proving this sum? $$\sum_{n=0}^N\binom{2N-n}N2^n(n+1)=(1+2N)\binom{2N}N$$ How to toggle or turn off text selection being sent to the clipboard I would like to disable the default behavior that seems to happen with every Linux distribution that I've ever tried that any selected text is immediately sent to the clipboard (Mint, Ubuntu, Fedora, SuSE, etc.) and every window manager (Gnome, KDE, Cinnamon), and instead to behave more like the Windows implementation.    I know that this is a beloved behavior by many in Linux, and I'm sure many will think I'm an idiot.  The reason I want to do this, is that I'm a keyboard junky when navigating a GUI.  (e.g.  when I'm in Linux and I copy a URL and then switch to my browser and type Ctrl+L, it selects the address bar and moves my intended paste down one notch and replaces it with what I'm trying to overwrite.)  I know there are MANY workarounds, but I don't really care about that, what I'd prefer is to be able to toggle the behavior for the clipboard. When editing a question on , I observed that a link which was not rendered in the post was hyperlinked in the edit preview (see pictures below). I tried reproducing it in the  to see if this also happens when the post is posted for the first time and observed the same. It seems to be a bug after the shift to CommonMark, leading to the different rendering of links on client and server-side. A link following a special character (without any space) is not hyperlinked in the post but is hyperlinked in the preview. A link separated from the special character with a white space appears to have no such problem. (see pictures below and my ). Link not hyperlinked in post  Same link hyperlinked in preview When is logistic regression solved in closed form? Reward for Passing Review tests? Just wanted to see what the community thought of adding a reward to passing the review tests? Maybe a 1 point reward?  I know seeing the tests in the review area has helped me be extra careful when reviewing, but I thought perhaps a little boost in incentive would be cool. Calculating the average of variably sized blocks of numbers within a column? Why does flag marking as helpful/declined not always correlate with moderator action?