Mathematics Questions, Problems & Solutions : TransWikia.com ~ Page 207

Prove that $sum_{n=1}^{infty }left ( frac {sin((2n-1)x)}{(2n-1)x)}right )^k frac{(-1)^{n-1}}{2n-1}=frac π 4$ for $0lt xlt frac pi {2k} $

Question :- Prove that$$sum_{ n =1}^{infty }left{frac{sinleft(left[2n - 1right]xright)}{left(2n - 1right)x}right}^{k}frac{left(-1right)^{n - 1}}{2n - 1} = frac{π}{4}qquadmbox{for}quad 0lt xlt frac{pi}{2k}$$While reading...

Asked on 08/13/2020 by Paras

1 answer

Vector field divergence equivalent definition

Could someone direct me to a proof showing the equivalence between the following two definitions of the divergence of vector field $F$ at $x$? (1) $lim_{|V| to...

Asked on 08/13/2020 by user_hello1

0 answer

Matrix representation of composition of linear transformations

I'm trying to know what is the matrix representation of a composition of linear transformations: $Vto E to W qquad T_1: TV=E quad T_2:TE=W $ Also $dim V=n qquad...

Asked on 08/12/2020 by Yudop

3 answer

Proof of The Third Isomorphism Theorem

Here's what I'm trying to prove right now: Let $V$ be a vector space over $mathbb{F}$. Let $M$ be a linear subspace of $V$ and ...

Asked on 08/12/2020 by Abhijeet Vats

1 answer

Fast paced but basic introduction to homological algebra

I'd like to learn some homological algebra. Being a physicist my abstract algebra background is not particularly strong so I find most of the usual books a bit forbidding. I...

Asked on 08/11/2020 by GFR

1 answer

Show estimator is consistent

For a special case of the gamma distribution: $$f(x)=frac{x}{theta^2}e^{-x/ theta}$$ $$E(x) = 2theta\ V(x) = 2theta^2$$ I find MLE of $hat{theta}$ to be $sum_{i=1}^n frac{x_i}{2n}$...

Asked on 08/08/2020 by M1996rg

1 answer

Counterexample of limsup of sets and measures

On page 163 in Wheeden-Zygmund, it is proved that for a nonnegative additive set function $phi$, $ overline{lim} phi(A_n) le phi(overline{lim} A_n)$ for any sequence of measurable functions...

Asked on 08/07/2020 by David Warren Katz

0 answer

mean distance between three point

I have a question that I would like your help solving. Let say I have three different points $a,b,c$ and a radius $r$. Let say the $A,B,C$...

Asked on 08/06/2020 by Yaniv765

0 answer

Character table of quaternions $Q_8$

I wanted to compute the character table of the quaternion group Q_8, and to this end I was reading this answer. For the degree 1 representation I used ...

Asked on 08/05/2020 by roi_saumon

1 answer

Evaluate $int frac{2-x^3}{(1+x^3)^{3/2}} dx$

Evaluate:$$int frac{2-x^3}{(1+x^3)^{3/2}} dx$$I could find the integral by setting it equal to $$frac{ax+b}{(1+x^3)^{1/2}}$$and differentiating both sides w.r.t.$x$ as$$frac{2-x^3}{(1+x^3)^{3/2}}=frac{a(1+x^3)^{3/2}-(1/2)(ax+b)3x^2(1+x^3)^{-1/2}}{(1+x^3)}$$$$=frac{a-ax^3/2-3bx^2}{(1+x^3)^{3/2}}$$Finally by setting ...

Asked on 08/05/2020 by Dharmendra Singh

4 answer

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