Comment by RandomGuy on Bounding above $a_{n+1}=\sqrt{16+|a_n|},\ a_0=4$
Or another way to think of it $x=\sqrt{16+\sqrt{16+\sqrt{...\infty}}}$ squaring on both sides you get $x^2=16+x$
View ArticleComment by RandomGuy on $f(X) = X^9 - 2dX^6 + 3d^2 X^3 - d^3$ irreducible in...
Did a more number than field theory proof
View ArticleComment by RandomGuy on Solving $\frac{dy}{dx}=\cos{x}\cos{y}$
what exactly do you mean by solution here cuz usually the function u found is the answer
View ArticleComment by RandomGuy on Need help with surface area and volume of "cheese slice"
is the outer edge circular or straight if it is circular is it the same radius?
View ArticleComment by RandomGuy on Stumped by simple change of variable in integration
We dont shift the function exactly shifting you add when you subtract is more like flipping and shifting we have two integration variables t and $u=(\frac{\pi}{2}-t)$ T is not including in the...
View ArticleComment by RandomGuy on Simplifying a Binomial Series
That is probability not number of ways that sum cannot converge to a fraction all its terms are integers
View ArticleComment by RandomGuy on Simplifying a Binomial Series
totally not an anagram of santa i assure you
View ArticleComment by RandomGuy on Simplifying $\frac{\tan(270 ^{\circ} -...
why without cot,sec,csc? that seems like an arbitrary restriction
View ArticleComment by RandomGuy on Six People Party (Proof Check)
Looks fine might want to use the pigeon hole principle to make it rigorous
View ArticleComment by RandomGuy on How to Prove n! is unbounded from above
Ah yes that would be proper mb
View ArticleComment by RandomGuy on $m_i, n_j$ integers and...
Is $k\neq k'$ else you could just take both sequences to be the same right?
View ArticleComment by RandomGuy on The Golden ellipse
i dont think "golden" figures are considered exclusive discoveries they just happen to have nice algebraic properties i could be wrong tho but i havent seen it treated specially in any books or such
View ArticleComment by RandomGuy on how to parameterize the boundary of a a parabolic...
possible repeat math.stackexchange.com/q/4843365
View ArticleAnswer by RandomGuy for How to prove the relation between Beta and Gamma...
In double integrals$$\int_c^d[\int_a^bf(x)g(y)dx]dy=\int_a^bf(x)dx\int_c^dg(y)dy$$So$$\begin{align}\Gamma(m)\Gamma(n)&=\int_0^\infty e^{-xy}x^ny^{n-1}\int_0^\infty...
View ArticleAnswer by RandomGuy for The problem of solve the limit separately
$$\lim_{n\rightarrow\infty} \left(n^{\frac{1}{n}}+\frac{3}{n}\right)^n=\lim_{n\rightarrow\infty} n\left(1+\frac{3}{n^{1+\frac{1}{n}}}\right)^n\ge \lim_{n\rightarrow\infty}n=\infty$$
View ArticleAnswer by RandomGuy for Given recursive formula $a_n = 2a_{n-1}+1$, show $a_n...
$a_n=2a_{n-1}+1$1.Characteristic equation$b_n:=a_n+1$$b_n-2b_{n-1}=0$The characteristic equation is now $x-2=0$So $b_n=2^nA \ \ \ a_n=2^nA-1$ and A is 1 from initial conditionsHow to get the...
View ArticleAnswer by RandomGuy for Solution verification - show that $a=b=c$ knowing an...
Since they are in proportion let k be a real number such that$$ax+b-ck=0\\bx+c-ak=0\\cx+a-bk=0\\$$Adding these three and cancelling (a+b+c) we get k=x+1 (which in in your solution)Now we get by...
View ArticleAnswer by RandomGuy for Value of $\sum_{k=0}^n \binom{n}{k}^2$ using analysis.
There is a way without differentiationLet $p(x)=(1+x)^n(1+\frac{1}{x})^n$Evidently $\sum{n \choose r}^2$ is the constant termRearranging we get $p(x)=\frac{(1+x)^{2n}}{x^n}$The constant term will be...
View ArticleAnswer by RandomGuy for Parametrise a Cylinder
You can't use a single parameter since single parameters give curves not surfacesThe closest thing I can think of iscylindrical coordinatesOr in other words parameterizing the circle and height...
View ArticleAnswer by RandomGuy for $\int \frac{\left(x+1\right)}{x\left(1+xe^x\right)^2}dx$
Use the substitution $t=xe^x \ \ \ \ dt=e^x(x+1)dx$$$\int \frac{e^x(x+1)}{xe^x(xe^x+1)^2}dx=\int \frac{dt}{t(t+1)^2}=\int...
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