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seen Jul 23 '13 at 15:47

Feb
24
answered Divergence of an improper integral - |cos(x^2)|/x^q
Feb
10
awarded  Citizen Patrol
Feb
9
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 $\lim_{x\to\infty}f(x)=\infty$.
Feb
5
accepted Question regarding Weierstrass M-test
Feb
5
comment Question regarding Weierstrass M-test
@Jonas: if you want to put that down as an answer, I'll gladly accept it.
Feb
5
comment Question regarding Weierstrass M-test
@Jonas: I see, something like $\sum \frac{(-1)^n}{n}$?
Feb
5
comment Question regarding Weierstrass M-test
@Jonas: not sure I follow. What should be constant?
Feb
5
comment Question regarding Weierstrass M-test
Thanks. One question though: why is $\sup_{x\in I} |u_n(x)| \geq u_n(n) = 1/n$?
Feb
5
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 $\frac{1}{n}$.
Feb
5
asked Question regarding Weierstrass M-test
Feb
4
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.
Feb
3
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 $[0, \infty)$
Feb
3
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.
Feb
3
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?
Feb
3
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 $f(x)\to 0^+$?
Feb
3
asked 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 $[0, \infty)$
Feb
1
comment Example of discontinuous $u(x)$ where $\sum u_{n}(x)\to u(x)$ uniformly in $I \subset \mathbb{R}$
$u_{n}(x)$ in $[0,1]$ doesn't go to $0$ here so it diverges but it can be fixed by taking $u_n(x) = \frac{1}{3^{n}}$ for example.
Jan
31
comment Example of discontinuous $u(x)$ where $\sum u_{n}(x)\to u(x)$ uniformly in $I \subset \mathbb{R}$
thanks, sorry for the confusion.
Jan
31
accepted Example of discontinuous $u(x)$ where $\sum u_{n}(x)\to u(x)$ uniformly in $I \subset \mathbb{R}$
Jan
31
revised Example of discontinuous $u(x)$ where $\sum u_{n}(x)\to u(x)$ uniformly in $I \subset \mathbb{R}$
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