Friday, 8 February 2019

abstract algebra - $[L_1 L_2 : k] = [L_1 : k] [L_2 : k]$ for two finite field extensions $L_1/k$ and $L_2/k$ with $L_1 cap L_2 = k$

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I want to prove or disprove the following: Let $L_1$ and $L_2$ be two finite extensions of field $k$ inside of an extension $L/k$. Moreo...

elementary set theory - Showing a function $H:mathbb R^{mathbb R}to mathbb R^{mathbb R}$ is onto and finding its inverse

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Let $H:\mathbb R^{\mathbb R}\to \mathbb R^{\mathbb R} \\ H(f)=\begin {cases}f^{-1} & \text {$f$ is a bijection} \\ f & else \end {...
Thursday, 7 February 2019

sequences and series - Evaluate $sum_{n=1}^infty frac{n}{2^n}$

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This is a homework question; I'm supposed to use power series to find the following sum: $$\sum_{n=1}^\infty \frac{n}{2^n}$$ I took the ...
Wednesday, 6 February 2019

real analysis - Is a periodic function differentiable? uniformly continuous?

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Let $f : \mathbb R \rightarrow \mathbb R$ be a function such that $f(x + 1) = f(x)$ for all $x \in \mathbb R.$ Which of the following statem...

modular arithmetic - GCD of two polynomials in Mod 2

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Let $p$ and $q$ be distinct primes. I wonder is the following statement always true? $$\gcd(x^p-1, x^q-1) \stackrel{?}{=} x-1$$ ...
Tuesday, 5 February 2019

integration - $int^infty_0frac{sin x}{x} , dx = frac{1}{2i}int^infty_{-infty} frac{e^{ix}-1}{x} , dx$, why?

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How comes this true? $$\int^\infty_0\frac{\sin x} x \, dx = \frac{1}{2i}\int^\infty_{-\infty} \frac{e^{ix}-1} x \, dx$$

sequences and series - What must be the simplest proof of the sum of first $n$ natural numbers?

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I was studying sequence and series and used the formula many times $$1+2+3+\cdots +n=\frac{n(n+1)}{2}$$ I want its proof. Thanks for...