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Evaluate$$prod^{100}_{k=1}left[1+2cos frac{2pi cdot 3^k}{3^{100}+1}right]$$My attempt: $$1+2cos 2theta= 1+2(1-2sin^2theta)=3-4sin^2theta$$ $$=frac{3sin theta-4sin^3theta}{sin theta}=frac{sin 3theta}{sin theta}$$ I did not understand how to solve after that. Help required....
Asked on 12/13/2020 by jacky
2 answerCan anyone give some hints about the following question?How many nonnegative integers $x_1, x_2, x_3, x_4$ satisfy $2x_1 + x_2 + x_3 + x_4 = n$?Normally this kind...
Asked on 12/13/2020 by Ivar the Boneless
3 answerWhile studing from Nakahara's book of Differential Geometry, I reached the part where topological spaces are introduced, and to define a topological space $mathcal{T}$, they take it as a...
Asked on 12/12/2020 by Jpmarulandas
1 answeri dont know how show that theorem is true Definition: A morphism $F: M longrightarrow N$ of $R$-modules is called regular if exist $G: N longrightarrow M$...
Asked on 12/12/2020 by Luiz Guilherme De Carvalho Lop
1 answerThe given rectangular equations are$$x^2+y^2+z^2=64$$$$(x-4)^2+y^2=16$$Converting to cylindrical coordinates I get$$r^2+z^2=64$$$$r=4cos(theta)$$So my triple integral is$$4int_0^{frac{pi}{2}}int_0^{4cos(theta)}int_0^{sqrt{16-r^2}}rdzdrdtheta$$It's a long and...
Asked on 12/12/2020 by Eric Brown
2 answerI am stuck with this question. How to prove that compliments of a universal set is the null set by contradiction?...
Asked on 12/12/2020 by JASDEEP SINGH
1 answerI and a friend are trying to find all endomorphisms $f$ of $mathcal{M}_n(mathbb{R})$ such that $f({}^t M)={}^t f(M)$ for all $M$. We believe they are of...
Asked on 12/12/2020 by Tuvasbien
2 answerWe know that 1/(1-x) = 1+x+x^2+x^3+....x^n Say we need to find a suitable function for the expansion x+x^2+x^3+x^4+....x^(n+1) We would differentiate 1/(1-x) = 1+x+x^2+x^3+....x^n This would yield 1/(1-x)^2 = ...
Asked on 12/12/2020 by dodgevipert56
1 answerBackground: I am trying to understand why, if $X$ is a random variable with $mathbb{E}big[|X|big]< +infty$ and $mu=mathbb{P}^{X}$, then $$frac{partial}{partial x_j}overline{mu}(u)=iint e^{i langle u,x rangle} x_j...
Asked on 12/12/2020 by Filippo Giovagnini
1 answerFrom my understanding of (smooth) manifolds, all you need is an atlas to describe a manifold. However, if you have some atlas ?={($U_n$,$phi_n$)} with $n$ charts, we...
Asked on 12/12/2020 by Tug Witt
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