daniel.jackson
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 Mar24 revised Question regarding positive-definite matrices changing transpose to transpose-conjugate Mar24 suggested approved edit on Question regarding positive-definite matrices Mar14 revised Regarding orthonormal basis added 188 characters in body Mar14 accepted Regarding orthonormal basis Mar13 asked Regarding orthonormal basis Feb24 answered Divergence of an improper integral - |cos(x^2)|/x^q Feb10 awarded Citizen Patrol Feb9 comment If $f(x)\nearrow\infty$ in $[a, \infty)$ and $f'(x)$ is continuous in $[a, \infty)$ then $\int_{a}^{\infty}\frac{f'(x)}{f(x)}\sin(f(x))$ converges @Adrian: I think it means $f(x)$ is increasing in the interval and $\lim_{x\to\infty}f(x)=\infty$. Feb5 accepted Question regarding Weierstrass M-test Feb5 comment Question regarding Weierstrass M-test @Jonas: if you want to put that down as an answer, I'll gladly accept it. Feb5 comment Question regarding Weierstrass M-test @Jonas: I see, something like $\sum \frac{(-1)^n}{n}$? Feb5 comment Question regarding Weierstrass M-test @Jonas: not sure I follow. What should be constant? Feb5 comment Question regarding Weierstrass M-test Thanks. One question though: why is $\sup_{x\in I} |u_n(x)| \geq u_n(n) = 1/n$? Feb5 comment Question regarding Weierstrass M-test @Jonas: interesting. Will $\log(1+x)=\sum (-1)^{n+1}\frac{x^n}{n}$ work? Here $I=(-1, 1]$ and at $x=1$ we'll get $\frac{1}{n}$. Feb5 asked Question regarding Weierstrass M-test Feb4 comment How to know that $a^3+b^3 = (a+b)(a^2-ab+b^2)$ I love how simple, yet very helpful your explanations are. Feb3 accepted Show $\int_{0}^{\infty}\sin(f(x))dx$ converges if $\int_{0}^{\infty}f(x)dx$ converges and $f(x)$ is a decreasing, continuous function in $[0, \infty)$ Feb3 comment Show $\int_{0}^{\infty}\sin(f(x))dx$ converges if $\int_{0}^{\infty}f(x)dx$ converges and $f(x)$ is a decreasing, continuous function in $[0, \infty)$ @Steven: could you give a hint as to why $f(x)\to 0$? I understand why intuitively but having some difficulty showing it... I'm familiar with a similar result that had $\lim_{x\to \infty}f(x)$ existed, then $\lim_{x\to \infty}f(x)=0$ must be true. Feb3 comment Show $\int_{0}^{\infty}\sin(f(x))dx$ converges if $\int_{0}^{\infty}f(x)dx$ converges and $f(x)$ is a decreasing, continuous function in $[0, \infty)$ @Joe: not sure I follow. You're saying $\int_0^\infty f(x) dx \implies \lim_{x\to \infty}f(x)=0$? What about $g(x)=x$ if $x \in N,\ g(x)=0$ otherwise? Feb3 comment Show $\int_{0}^{\infty}\sin(f(x))dx$ converges if $\int_{0}^{\infty}f(x)dx$ converges and $f(x)$ is a decreasing, continuous function in $[0, \infty)$ @Arturo: Why does $f(x)\to 0^+$?