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Find for which $alpha$ the integral $int_{0}^{1} frac{1-x^{alpha}}{1-x}dx$ converges. My Attempt: suppose $f(x) = frac{1-x^{alpha}}{1-x}$. I think that the integral converges for $alpha > -1$. First...
Asked on 11/09/2021
3 answerShow that the following power series satisfies this functional equation $$fleft(dfrac{2x}{1+x^2}right)=(1+x^2)f(x),,$$where the series given is $$f(x)= 1+dfrac{1}{3}x^2+dfrac{1}{5}x^4+dfrac{1}{7}x^6+ cdots,.$$I can painstakingly get a relation between the derivatives such...
Asked on 11/09/2021 by crystal_math
3 answerHello I am a high school student from germany and I am starting to study math this october. I am trying to prepare myself for the analysis class which I...
Asked on 11/09/2021 by Bill Matter
3 answerI'm reading Atiyah's article about his homonymous theorem but cannot figure out some of the notations he used. I'm not very familiar with algebraic geometry, so I thought I could...
Asked on 11/09/2021 by Fede96
0 answerI was given the above question to answer, and I do not know how to do it. If t is...
Asked on 11/09/2021
2 answer$S$ is any set of a finite-dimensional inner product space $V$. Prove or disprove that $S$ is a subspace of $V$ iff $(S^{perp})^{perp}=S$.My approach was...
Asked on 11/09/2021
2 answerLet $X$ be a set. How many $sigma$-algebras of subsets of $X$ contain exactly $m$ elements? Any hints for how to begin a solution to this...
Asked on 11/09/2021 by user1333549
1 answerI want to simplify the following expression: $frac{{left( {1 + sqrt 3 tan {1^{circ}}} right)left( {1 + sqrt 3 tan {2^{circ}}} right)left( {tan {1^{circ}} + tan {{59}^{circ}}} right)left( {tan...
Asked on 11/09/2021
2 answerLet $mathbf{v}inmathbb{C}^n$ with $v_k$ as its $k^{text{th}}$ entry. We know that $lVertmathbf{v}rVert^2=underset{k=1}{overset{n}{sum}}|v_k|^2$ is the square of the $L^2$ norm of $mathbf{v}$. I wonder, does...
Asked on 11/09/2021
0 answerLet $f$ analytic in $D=left{z:|z|<1right}$ such that $f(0)=0$ and $|f(z)|leq |z+3/2|$ for all $zin D$.Show that: (a) $|f(1/2)|leq 1$ (b) Find all functions...
Asked on 11/09/2021
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