Vandermonde's Convolution for odd terms
Combinatorics That Looks Similar to Vandermonde's Identity
For which $n$ is the sum: $$\sum_{k=0}^{n}10^k$$ a prime number? Are they finite? One of my laptop's keys has fallen off. Is there any way I can remap another key to serve as that key?  I'll 'sacrifice' the other key because I never use it. Derivation of $E=pc$ for a massless particle? In classical mechanics, massless particles don't exist because for $m=0$, $p=0$.  The relativistic relation between energy, mass and spatial momentum is: $E^2= (pc)^2 + (mc^2)^2$ . So it is said that setting $m=0$ in the first equation you get $E=pc$.  How could setting $m=0$ in that equation give you $E=pc$ whilst $p$ appears in the equation and we know $p=γmu$? If you set $m=0$ you will have indeterminacy due to "$γm$". It seems to me like we are doing a "trick" in order to get the $E=pc$. Perhaps there is another proof for this relation? After taking a statistics course and then trying to help fellow students, I noticed one subject that inspires much head-desk banging is interpreting the results of statistical hypothesis tests.  It seems that students easily learn how to perform the calculations required by a given test but get hung up on interpreting the results.  Many computerized tools report test results in terms of "p values" or "t values".  How would you explain the following points to college students taking their first course in statistics:   What does a "p-value" mean in relation to the hypothesis being tested?  Are there cases when one should be looking for a high p-value or a low p-value? What is the relationship between a p-value and a t-value? work done is zero if displacement of point of application of force is zero but why can't we assume forces on COM and its displacement I am trying to build a computer, placing an  in a custom .  The case is an extra small form factor which can accommodate the Mini ITX board, but it requires an external power supply, similar to the power-bricks that laptops use.  I am looking at different power supplies and like , but I cannot figure out whether/how to power this system, because they provide a 24-pin connector, but the motherboard has a 4-pin connector as well.  Is the 4-pin connector optional or is it required? If it is required, how can I provide power for this system? I have a problem understanding the proof of Rencontres numbers (Derangements) I understand the whole concept of Rencontres numbers but I can't understand how to prove this equation  $$D_{n,0}=\left[\frac{n!}{e}\right]$$  where $[\cdot]$ denotes the rounding function (i.e., $[x]$ is the integer nearest to $x$). This equation that I wrote comes from solving the following recursion, but I don't understand how exactly the author calculated this recursion.  $$\begin {align*}  D_{n+2,0} &amp; =(n+1)(D_{n+1,0}+D_{n,0}) \\ D_{0,0} &amp; = 1 \\ D_{1,0} &amp; = 0  \end {align*} $$ Why do carnivores evolve? If humans can live without consuming other animals, then why do we do it? From a biological point of view, why do we eat meat? I would also extend the question to other animals because many animals are carnivorous: why did carnivory evolve and why do animals eat meat? How to remove all white spaces just between brackets [] using Unix tools? LEFT JOIN vs. LEFT OUTER JOIN in SQL Server What is the difference between LEFT JOIN and LEFT OUTER JOIN? how can I find sum of series $ \sum_{n=1}^{\infty} (n\cdot \ln \frac{2n+1}{2n-1} - 1) $? It is so weird for me because I put this to Mathematica and it tells me that sum does not converge...  Let consider sum no to infinity, but to n  $$ \sum_{k=1}^{n} (k\cdot \ln \frac{2k+1}{2k-1} - 1) =$$ $$ ln \frac{3}{1}\cdot \left(\frac{5}{3}\right)^2 \cdot...\cdot \left(\frac{2n+1}{2n-1}\right)^n - n = ln \frac{1}{1}\cdot \frac{1}{3}\cdot \frac{1}{5}\cdot ... \frac{1}{2n-1} - n $$ but $$ n = ln e^n $$ so it will be $$ln\frac{1}{e^n} \cdot \frac{1}{1}\cdot \frac{1}{3}\cdot \frac{1}{5}\cdot ... \frac{1}{2n-1}$$  So the limit of it is $-\infty$ Have I done this well or I missed sth? Why are pointers such a leading factor of confusion for many new, and even old, college level students in C or C++?  Are there any tools or thought processes that helped you understand how pointers work at the variable, function, and beyond level?  What are some good practice things that can be done to bring somebody to the level of, "Ah-hah, I got it," without getting them bogged down in the overall concept? Basically, drill like scenarios. Where did Magister Illyrio Mopatis get the dragon eggs from? When to use an object pronoun or a possessive adjective before a gerund It is easily checked that $\displaystyle\sum_{i\ =\ 0}^{n}\left(\, -1\,\right)^{i} \binom{n}{i} = 0$, for example by appealing to the binomial theorem.  I'm trying to figure out what happens with the truncated sum $\displaystyle\sum_{i\ =\ 0}^{D}\left(\, -1\,\right)^{i}\binom{n}{i}$.  How far away from $0$ can this get, as a function of $D$ ?.  I'm mostly interested in the case of when $D \ll n$, such as $D \sim \,\sqrt{\,n\,}\,$.  Thanks ! For context, see .  In essence:     "This question is protected to prevent "thanks!", "me too!", or spam answers by new users. To answer it, you must have earned at least 10 reputation (not including the ) on this site."   Suggested addition to existing message is in bold. Why is propagation of uncertainties quadratic rather than linear? 1) Up until now, during practical work sessions, I always used these formulas for uncertainty propagation:  if $C = A+B$  or  $C = A-B$ $$\Delta C = \Delta A + \Delta B$$ if $C = AB$ or $C = \frac{A}{B}$ $$\frac{\Delta C}{C} = \frac{\Delta A}{A}  + \frac{\Delta B}{B}$$ if $C = A^m$ $$\frac{\Delta C}{C} = |m|\frac{\Delta A}{A}$$   These formulas are derived from the expression for the differential of a function of multiple variables:  if $C = f(A,B)$ $$ dC = \frac{\partial C}{\partial A}dA + \frac{\partial C}{\partial B}dB \Rightarrow \Delta C = \frac{\partial C}{\partial A}\Delta A + \frac{\partial C}{\partial B}\Delta B$$  and I think that makes sense because the value C is just a function of two variables that happen to be measurements.  2) But this morning, I was told that this is wrong and I should actually use these instead:  if $C = A+B$  or  $C = A-B$ $$(\Delta C)^2 = (\Delta A)^2 + (\Delta B)^2$$ if $C = AB$ or $C = \frac{A}{B}$ $$\left(\frac{\Delta C}{C}\right)^2 = \left(\frac{\Delta A}{A}\right)^2  + \left(\frac{\Delta B}{B}\right)^2$$  which are apparently derived from a general formula:  $$(\Delta C)^2  = \left(\frac{\partial C}{\partial x_1}\Delta x_1\right)^2 + \left(\frac{\partial C}{\partial x_2}\Delta x_2\right)^2 +   \hspace{0.3cm}...$$  or   $$\Delta C  = \sqrt{\left(\frac{\partial C}{\partial x_1}\Delta x_1\right)^2 + \left(\frac{\partial C}{\partial x_2}\Delta x_2\right)^2 +   \hspace{0.3cm}...}$$  3) So:   Why is this formula better?  Where does it come from? What does it actually represent? (Do I recognize the shape of a norm in that last formula?)  What is wrong with the formula of the differential?   I'm asking a lot, but the propagation of uncertainty always confused the ship out of me. When (and why) should you take the log of a distribution (of numbers)? What is the correct position of the word 'also'?     I would like to also talk about ...      I would also like to talk about ... The observable universe is approximately 13.7 billion years old. But yet it is 80 billion light years across. Isn't this a contradiction?