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Oct
26
revised Limit of $x_n/n$ as $n\to\infty$
added 76 characters in body
Oct
25
comment Limit of $x_n/n$ as $n\to\infty$
From my answer you can see that letting $y_n=x_n^2$ is a good choice for your question (1) but not as good a choice for your question (2). The reason for letting $y_n=x_n^2$ in question (1) can be understood in this way: the associated function $f$ is question (1) is $c x^2$ for some $c>0$. For the same reason, in question (2), a good choice is letting $y_n=x_n^{3/2}$.
Oct
25
revised Limit of $x_n/n$ as $n\to\infty$
added 2340 characters in body
Oct
23
revised Limit of $x_n/n$ as $n\to\infty$
added 141 characters in body
Oct
23
answered Limit of $x_n/n$ as $n\to\infty$
Oct
22
reviewed Leave Open Calculating index of a subgroup
Oct
22
comment Is the image of a nowhere dense closed subset of $[0,1]$ under a differentiable map still nowhere dense?
@NateEldredge: Here $E$ is closed and hence....
Oct
22
comment A Problem on Improper Integrals
@ParamanandSingh: Thank you for your tolerance.
Oct
22
comment A Problem on Improper Integrals
@ParamanandSingh: You are welcome! I still feel sorry about misleading you with a flawed argument. Maybe I I shouldn't post any answer when my head is not clear enough. Don't worry about the bounty; you can wait until the bounty expires to see if there is a better answer.
Oct
22
comment A Problem on Improper Integrals
@ziangchen: Thank you again for your appreciation.
Oct
22
comment A Problem on Improper Integrals
@ParamanandSingh: Terribly sorry! I mistakenly thought $h(0)=0$ before. I have updated my answer.
Oct
22
comment A Problem on Improper Integrals
@ziangchen: No, it was flawed, and now it's corrected. My brain is a little unclear today and I mistakenly thought $h(0)=0$ before. Thank you!
Oct
22
revised A Problem on Improper Integrals
added 54 characters in body
Oct
22
comment A Problem on Improper Integrals
@ParamanandSingh: I wrote my answer in this way because I thought you would prefer a hint rather than a full answer. Is it better if I add all the details to my answer?
Oct
22
comment For a closed plane curve, showing some inequalities.
@JeongNam-ho: You are welcome.
Oct
22
revised A Problem on Improper Integrals
added 150 characters in body
Oct
22
comment For a closed plane curve, showing some inequalities.
@JeongNam-ho: You don't have to hear that. You can prove it by yourself, as I said in the first comment.
Oct
22
answered A Problem on Improper Integrals
Oct
22
comment For a closed plane curve, showing some inequalities.
@JeongNam-ho: Yes, $\langle,\rangle$ denotes inner product. I thought it was a conventional notation, so I didn't mention it. Sorry about the confusion. For the remark, I think it's not very hard to prove, but it need quite a few words to write down all the details.
Oct
22
comment For a closed plane curve, showing some inequalities.
@JeongNam-ho: It's just the linearity of integral. To see this more clearly, you may write write $v=(v_1,v_2)$, $\gamma'(t)=(x'(t),y'(t))$, and evaluate both sides of the equality.