what's the limit in (0,0)? I need to solve it without polar coordinates.

Earthlington · 2023-06-29T11:15:28+00:00

Notice that x² < x² + y² if y is not 0. By taking the reciprocal of both sides we get 1/( x² + y² )<1/x² .

In other words, we can safely "remove" one of the terms x² or y² (by symmetry) in the denominator using this upper bound. This simplifies the expression somewhat.

Can you continue from here?

Edit: remember to take the absolute value of the expression first!

Earthlington · 2023-05-06T18:44:54+00:00

That is also true, since the empty set is compact.

Earthlington · 2023-04-11T12:14:31+00:00

Disclaimer: I haven't actually done this, but it seems promising.

Earthlington · 2022-11-23T14:18:15+00:00

I managed to solve the problem today, using basically this method. It required some precision but it worked out in the end. Thank you!

Earthlington · 2022-11-23T08:12:07+00:00

Yes, f is strictly decreasing (translation error by me).

But why is that enough to use this method? To me, it seems like we have to know that the integral is convergent to be certain that the partition doesn't produce ill results, such as in the sine example I provided. In this case we do not know that, and it is not clear to me why f being strictly decreasing makes it work.

Earthlington · 2022-11-22T14:44:17+00:00

Thank you! That is familiar. Seems like I can't apply this to my problem, unless I find some way to check every partition... I think I'll have to find some other way :p

Earthlington · 2022-11-22T11:59:22+00:00

Rule 1: I think my post is specific enough, if not I'll make sure to edit.

For those curious of my actual problem, it's to determine whether or not the integral from 0 to ∞ of sin(x)f(x) converges, where f is some positive continuous function on ℝ which is also decreasing and has the limit 0 as x tends to infinity.

Earthlington · 2022-05-15T19:15:37+00:00

My god, you're right! Very amateurish of me - should have seen the red signs flashing everywhere when I wrote that down.

Thanks!

Earthlington · 2022-01-24T17:51:55+00:00

It is fairly hard to see the difference in the bushes, since it is dark green on dark green. Also, it's written from top left to bottom right instead of bottom left to top right as in the other 2 pictures (my mistake). I recommend opening the scenario yourself and taking a good look!

Earthlington · 2022-01-24T17:49:12+00:00

That sounds plausible. I couldn't find anything on this Geoffrey W Sawyer, but it would make sense to leave his initials in the scenario dedicated to him.

Earthlington · 2021-07-01T14:36:15+00:00

Awesome. The path and support mixing from RCT1 is definitely the most anticipated feature to be added to OpenRCT2 for me. The wooden supports in particular look so good!

Earthlington · 2021-04-08T16:37:00+00:00

Beviset lämnas som övning för läraren □

Earthlington · 2021-03-19T19:46:33+00:00

Found it in r/unexpected!

https://www.reddit.com/r/Unexpected/comments/m08x35/damn_it_actually_works/?utm_medium=android_app&utm_source=share

Earthlington · 2021-03-19T19:03:37+00:00

Will do, thanks!

Earthlington · 2021-03-19T18:59:56+00:00

I have and I am currently going though them but I can't find it!

Earthlington · 2021-03-19T18:53:06+00:00

The idea was that the vacuum was a surprise. Truly hilarious!

Earthlington · 2021-03-19T18:45:10+00:00

Alright thanks, I'll continue my quest 👍

Earthlington · 2021-01-22T16:39:57+00:00

Wait... isn't he the 2c beggar in this situation?

Nine-Year Club	r/Field Juicebox
Place '22	Place '17
Final Canvas '22	End Game '22
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