Evaluating $\sum _{n=2}^{\infty } \frac{(-5)^n}{8^{2 n}}$ making use of geometric series
Evaluating $\sum_{n=1}^\infty \frac{2^{2n-1}}{5^{n+1}}$
Find all functions $f\colon \mathbb R\rightarrow \mathbb R$ such that for all reals $x$ and $y$:   $$f(f(x)f(y))+f(x+y)=f(xy).$$   It was six hours ago in IMO 2017 (problem 2).  I tried  the standard way: $x=0$, $x=y$, $x=1$,etc. but without any success.   Let $f(0)=a$. Hence, for all $x\in\mathbb R$ we have $f(a(f(x))+f(x)=a$. What is the rest?   Also we have $(x-1)(y-1)-1+x+y-1=xy-1$ and  $1-(1-x)(1-y)+1-x-y=1-xy$. Anthology series with suspended animation and a galaxy-wide empire? Prove that the square root of a positive integer is either an integer or irrational Is my proof that the square root of a positive integer is either an integer or an irrational number correct?  The proof goes like this:  Suppose an arbitrary number n, where n is non-negative. If $\sqrt{n}$ is an integer, then $\sqrt{n}$ must be rational. Since $\sqrt{n}$ is an integer, we can conclude that n is a square number, that is  for some integer a.  Therefore, if n is a square number, then $\sqrt{n}$ is rational.  Suppose now that n is not a square number, we want to show that the square root of any non-square number is irrational.  We prove by contradiction. That is, we suppose that the square root of any non-square number is rational. So $\sqrt{n} = \frac{a}{b}$, where $a,b \in Z^+, b \neq 0$. We also suppose that $a \neq 0$, otherwise $\frac ab = 0$ , and n will be a square number, which is rational.   Hence $n = \frac {a^2}{b^2}$, so $nb^2 = a^2$.    Suppose $b=1$. Then $\sqrt n = a$ , which shows that n is a square number. So $b \neq 1$. Since $\sqrt n &gt; 1$, then $a&gt;b&gt;1$.  By the unique factorization of integers theorem, every positive integer greater than $1$ can be expressed as the product of its primes.  Therefore, we can write $a$ as a product of primes and for every prime number that exists in $a$, there will be an even number of primes in $a^2$. Similarly, we can express $b$ as a product of primes and for every prime number that exists in $b$, there will be an even number of primes in $b^2$.  However, we can also express $n$ as a product of primes. Since $n$ is not a square number, then there exist at least one prime number that has an odd number of primes.  Therefore, there exists at least one prime in the product of $nb^2$  that has an odd number of primes. Since $nb^2=a^2$ , then this contradicts the fact that there is an even number of primes in $a^2$ since a number can neither be even and odd.  Therefore, this contradicts the fact that $\sqrt n$ is rational. Therefore, $\sqrt n$ must be irrational.  Is this sufficient? Or is there any parts I did not explain well? If $E$ has Lebesgue measure $0$, must there exist a translate such that $E\cap E+x=\varnothing$? If $y\in E\cap E+x$ then $y=h=h'+x \implies x=h-h'$ for $h,h'\in E$. Likewise, if $x=h-h'$, then $h\in E\cap E+x$. So that $E\cap E+x$ is non empty iff $x=h-h'$ where $h,h'\in E$. Thus, if there exists a real $x$ which cannot be expressed as above, the result is proven. The problem is I cannot prove that such an $x$ must exist. I have tried assuming that all real $x$ are of that form, but that gives me nothing except that $E$ must be uncountable and unbounded. Any hints? How can I make a PATCH HTTP Callout from Apex? I need to make a PATCH http request.  I made the rather incredible discovery today that I can't set an HttpRequest to use the PATCH method, as when sending the request I get a callout exception:     System.CalloutException "Invalid HTTP method: PATCH"   The work around I've found on Stack Exchange and elsewhere is only applicable to a Salesforce Rest API request:    This is therefore not helpful to me as I'm calling out to a third party.  By what manner of magic can I send a PATCH request from Salesforce?  The only work around I can think of is to not do it from Apex - this is not an option. White's test for heteroskedasticity in R I am trying to estimate heteroskedasticity in R. I had Eviews available in my college's lab but not at home. I have been trying to use "het.test" package and whites.htest but the value that I get is different from what I get in Eviews. According to the author of the package, it is meant to do the same test as is done in Eviews.   My model looks like A~B+C. Kindly guide me what I am doing wrong.   I 1st run ols for equation A~B+C, after getting results, I clicked View&gt;Residual diagnostics&gt;Heteroscadicity tests&gt;White's (with Include white cross terms) and then Press ok. The result i got was Prob. F(5,27) = 0.2948.   For R, i tried code at this link [this link][1], here is the edited code that i used.  library(vars)  library(het.test) dataset &lt;- data.frame(A,B,C))  model1 &lt;- VAR(dataset, p = 1)  whites.htest(model1)   This gave me value of 0.63.  I tried searching this forum but none question seemed related. I am not a programmer not statistician so please pardon my ignorance. Am I able to use a controller to play the PC version of Minecraft? If so, can I just plug it in and play or do I have to do something else before being able use one? Does the  core rulebook contain any information that is not included already on  that would make it worth it to buy the PDF version of the manual? Or does it only have extra descriptive text? Render result is completely blank What is meant by "Constant Amortized Time" when talking about time complexity of an algorithm? Why on average does each bootstrap sample contain roughly two thirds of observations? Setting time limit php In Blender 2.8, how do I light an EEVEE scene with an HDR? I installed Ubuntu 13.04 on my Vista 64 bit machine and I thought it was going to uninstall Vista but it didn't, now I have both systems.  How do I get rid of Vista? How is light affected by gravity? Light is clearly affected by gravity, just think about a black hole, but light supposedly has no mass and gravity only affects objects with mass.  On the other hand, if light does have mass then doesn't mass become infinitely larger the closer to the speed of light an object travels. So this would result in light have an infinite mass which is impossible.  Any explanations? How to make a spiked sphere? I know how to make it manually with cone rotation around the sphere. but maybe there's much simpler solution or addon that has that option, preferably not very high poly solution.  Also if possible I would like the spikes not to be very pointy if that makes any sense      Thanks Proof of Young's inequality The following problem is from Spivak's Calculus.  Suppose that $f$ is a continuous increasing function with $f(0)=0$. Prove that for $a,b \gt 0$ we have Young's inequality   $$ ab \le \int_0^af(x)dx+\int_0^bf^{-1}(x)dx$$, and that equality holds if and only if $b=f(a)$.  It is enough to consider the case $f(a) \gt b$, and show that the strict inequality occurs in this case.  I've tried proving this using the theorem  $$ \int_a^bf^{-1}=bf^{-1}(b)-af^{-1}(a)-\int_{f^{-1}(a)}^{f^{-1}(b)}f$$  but I got stuck along the way.   How may I show this rigorously using the definition or properties of integrals? Any hint, suggestions or solutions would be appreciated. In reading through the original GCM specification (), the composition of the 128 bit Initialization Vector as a concatenation of a 96b nonce and a 32b unsigned wrapping counter seems arbitrary and forces the scrambling pattern to repeat every 232 16-byte block.  Is the algorithm expected to be secure for significantly longer stream lengths if for example the IV were a 128b nonce XORed with a 64b or 128b counter, or are there known cryptanalysis issues that begin to arise? I have heard that two permutations are conjugate if they have the same cyclic structure. Is there an intuitive way to understand why this is? There are many beings that have very big powers. There are beings that are worshipped as gods (Asgardians; and I would lump Azazel into that as well since Marvel seems to imply that he's - while a mutant - someone people consider to be "Devil"). There are mutants whose power would clearly have been considered as "god like" by people worshipping Greek or other polytheistic pantheons where the threshold to what made you a god was pretty low.  But are there any actual major "Gods" in Marvel universe?  Since there are many different definition of what a "god" is, the one that is pertinent to this question:   Power wise, being able to create Universe(s); e.g. gods more in a sense of Christianity/Judaism scale rather than "Pan" or "Hestia" in Greek pantheon who merely have some limited supernatural powers over small domain of natural events. Effectively immortal, at least excluding specific exotic attacks. Origin-wise, be "supernatural", not merely a human mutant or a powerful alien with advanced technology. Desire/demand/like followers' worship or prayers.