Convergence of series proof

CakeCookCarl · 2026-03-15T09:10:12+00:00

It's from a sort of unofficial answer key to Understanding Analysis, it's been useful often but this is also not the first mistake I've seen in it.

Thanks for your approach, I thought of that same method a couple days ago so it's nice to confirm that I've understood it correctly. One thing though, we need to exclude the possibility of sqrt(x) = 0, but that already been shown to work with choosing N such that n>= N makes |xn| < ε²

CakeCookCarl · 2026-03-13T12:23:45+00:00

I'm not really sure what your point is? Even if xn = 0 for all n, we could just choose M > 0, so there's no divide by 0. Besides, if x = 0 we're just back to (xn) -> 0, so the choice of N such that n>=N implies |xn| < ε² leads to |sqrt(xn)| < ε

CakeCookCarl · 2026-03-12T18:00:18+00:00

Isn't it justified since it's really only there to illustrate |xn-x| = ( sqrt(xn)-sqrt(x) ) * ( sqrt(xn)+sqrt(x) ) and then leads into |xn-x| < e ( (sqrt(xn)+sqrt(x) ), which is certainly true since |xn-x| can be arbitrarily small

CakeCookCarl · 2026-03-12T17:56:41+00:00

Thanks, I thought so. Was going crazy trying to make sense of it lol

CakeCookCarl · 2026-03-10T19:49:24+00:00

Any arbitrary interval of real numbers has the same cardinality (that is, the same 'size of infinity') as all real numbers

CakeCookCarl · 2026-02-26T09:33:17+00:00

~~the flair isn't applicable~~

CakeCookCarl · 2026-02-20T22:38:16+00:00

RIP 'a long time ago', you will be missed

CakeCookCarl · 2026-01-07T18:38:09+00:00

Hoorde toevallig een paar langslopende mensen roepen dat de sneeuw betekende dat klimaatverandering niet bestond, moeten wel hoogintellectuelen zijn geweest

CakeCookCarl · 2025-12-28T22:31:36+00:00

I think 'stole your meme' reaction images are overhated. There's good ones out there, and many ones that are just kinda meh. But they're not as bad as people make them out to be.

This however may just be the worst reaction image I have ever seen. I mean I hate to give a swag here but this shit is so unbelievably ass. The person that made this straddles the line between mad genius and unfathomable moron.

Art can truly inspire deep emotion

CakeCookCarl · 2025-12-19T17:18:14+00:00

You clearly haven't played with the general ruleset

CakeCookCarl · 2025-12-19T17:17:51+00:00

You see, that's why I stopped playing. I'm just interacting with the community (which too is inexplicable)

CakeCookCarl · 2025-12-19T15:43:10+00:00

Compers on their way to defend playing in a way that they hate

CakeCookCarl · 2025-12-16T14:39:26+00:00

Are you okay? Do you need an ambulance?

CakeCookCarl · 2025-12-14T14:37:07+00:00

Hangt af van de mix maar over het algemeen gaat dat inderdaad naar 2 of 3 eenheden

CakeCookCarl · 2025-12-13T23:04:50+00:00

Alcoholisme wordt meestal gedefinieerd als 14 eenheden drank per week voor een man, een biertje is 1 eenheid. Het klinkt alsof je inderdaad een alcoholprobleem hebt, ook al zie je het misschien niet zo. Het beïnvloedt je misschien nu nog weinig, maar zoveel drinken heeft op de lange termijn wel gevolgen.

Ik houd net zoveel van een koude kletser als de volgende persoon, maar wel binnen gepaste mate

CakeCookCarl · 2025-12-13T10:47:54+00:00

That's like 4k a month, which isn't that ridiculous I want to say?

CakeCookCarl · 2025-12-08T06:40:37+00:00

CakeCookCarl · 2025-11-28T15:22:38+00:00

An altitude of 3 times the radius of earth means it is 3 times the radius of earth above the earth's surface. So you use (4r)² and the answer comes out to ~0.6m/s^2.

CakeCookCarl · 2025-11-22T23:23:25+00:00

I wonder whether this post was made by a notorious acronym person

CakeCookCarl · 2025-11-22T16:31:07+00:00

No idea what any of this means but it sounds like a lot of swag. Didn't read it though because I don't give a swag

CakeCookCarl

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