Anonymous at Tue, 2 Jul 2024 15:47:41 UTC No. 16264410

>>16264395
we do it this way because these properties are eventually necessary and useful at some point or another.

if you had [math]\epsilon > 0[/math] s.t. [math]\not\exists N[/math] s.t. [math]n>N\implies|a_n-L| < \epsilon[/math], then you'd have a sequence with no limit and you'd only need to find one [math]\epsilon[/math] to prove a sequence doesn't have a limit.

Anonymous at Tue, 2 Jul 2024 16:03:23 UTC No. 16264431

>>16264410
Latexspeak works in 4chan?

Anonymous at Tue, 2 Jul 2024 16:05:19 UTC No. 16264435

>>16264395
how is this mumbo jumbo?

Anonymous at Tue, 2 Jul 2024 16:06:52 UTC No. 16264436

>>16264395
That thing explains the convergence of a sequence dumbass

Anonymous at Tue, 2 Jul 2024 16:15:01 UTC No. 16264447

>>16264395
Freshman year must've been tough huh?

Anonymous at Tue, 2 Jul 2024 17:00:12 UTC No. 16264512

>infinitely often
>almost surely
Is it only me or do those two terms sound like they say the same thing? I know they're different concepts but the words are so unevocative to me they prevent me from getting a good intuition.

Anonymous at Tue, 2 Jul 2024 17:27:43 UTC No. 16264550

i keep shitting and cumming because my prof wont release summer grades. wtf kind of school doesnt require grades posted within a week of finals? hey TAs, fuck you and suck on the pp lmao

Anonymous at Tue, 2 Jul 2024 17:36:09 UTC No. 16264565

>>16264512
I'll take it you're referring to "almost surely infinitely often" as it describes the event [math]A_n[/math], with probability [math]P(\lim{\sup{A_n}})=1[/math]. This is where [eqn]\lim{\sup{A_n}}=\bigcap\limits_{m\leq{1}}\bigcup\limits_{n\leq{m}}A_n.[/eqn] You should be aware that this is, by definition, the event where infinitely many [math]A[/math] events occur -- thus the infinitely often. As its probability is 1 (out of the maximum 1), the chances of this event occurring infinitely often becomes (almost) surely.

Contrast this against "almost surely finitely often", which is the case where the chances of the infinite-event case are 0. The event [math]A_n[/math] then occurs finitely often almost surely.

Does that make sense?

Anonymous at Tue, 2 Jul 2024 17:46:45 UTC No. 16264581

>another freshman filtered by the definition of a limit

Anonymous at Tue, 2 Jul 2024 18:03:49 UTC No. 16264608

>>16264581
for the life of me, I cannot remember. what made us think we're wise and we'd never compromise

Anonymous at Tue, 2 Jul 2024 19:12:28 UTC No. 16264744

>>16264395
infinity isnt defined at all here so its an incomplete definition.

Anonymous at Tue, 2 Jul 2024 21:01:19 UTC No. 16264931

Imagine getting filtered by this
Man you should really start trade or finance. I can bet that in couple of years you'll be rich than most of us.

Anonymous at Tue, 2 Jul 2024 21:01:53 UTC No. 16264934

>>16264431
Yes, you can use [math] with the corresponding closing bracket /math

Anonymous at Tue, 2 Jul 2024 21:12:46 UTC No. 16264957

>>16264608
Based millennial

Anonymous at Wed, 3 Jul 2024 07:26:17 UTC No. 16265606

>>16264934
\in \cdots

Anonymous at Wed, 3 Jul 2024 07:27:33 UTC No. 16265608

>>16265606
$\in \cdots$

Anonymous at Wed, 3 Jul 2024 07:29:01 UTC No. 16265610

>>16265608
doesn't work

Anonymous at Wed, 3 Jul 2024 07:29:28 UTC No. 16265611

>>16265606

Anonymous at Wed, 3 Jul 2024 07:32:10 UTC No. 16265614

>>16265611
[math]\in\cdots[/math]

Anonymous at Wed, 3 Jul 2024 07:34:05 UTC No. 16265615

>>16265614
[math]\cdots\in[/math]

Anonymous at Wed, 3 Jul 2024 07:37:11 UTC No. 16265616

[math] \displaystyle
f(x) = e^{-ix}(\cos x + i \sin x)
\\
f^{\prime}(x) = e^{-i x}(i \cos x - \sin x) - i e^{-i x}(\cos x + i \sin x)
\\
f^{\prime}(x) = e^{-i x}(i \cos x - \sin x) - e^{-i x}(i \cos x + i^2 \sin x) \equiv 0
\\
f^{\prime}(x) = 0 \;\;\; \forall \; x \in \mathbb{R}\Rightarrow f(x) \text{ is a constant}
\\
f(0) = e^{0}(\cos 0 + i \sin 0) = 1 \cdot(1+0) = 1 \Rightarrow f(x) = 1 \;\;\; \forall \; x \in \mathbb{R}
\\ \\
1 = e^{-ix}(\cos x + i \sin x) \Rightarrow e^{ix}=\cos x + i \sin x \;\;\; \forall \; x \in \mathbb{R}
[/math]

🗑️ Anonymous at Wed, 3 Jul 2024 07:38:43 UTC No. 16265617

laplace transformation from power series
[math] \displaystyle
\sum_{n=0}^{\infty} a_n x^n = A(x) \\
\sum_{n=0}^{\infty} a(n) x^n = A(x) \\
a(n) \sim \! \sim \! \sim \! \! \! > A(x) \\
a(n) = 1 \sim \! \sim \! \sim \! \! \! > \frac{1}{1-x} \: \: \, ,\: \: \: \left | x \right | < 1 \\
a(n) = \frac{1}{n!} \sim \! \sim \! \sim \! \! \! > e^x \\
\text{continuous analog:} \\
\int_{0}^{\infty} a(t) x^t dt = A(x) \: \: \, ,\: \: \: 0 < x < 1 \Rightarrow ln(x) < 0 \\
x^t = \left ( e^{ln(x)} \right )^t \: \: \, ,\: \: \: -s = ln(x) \Rightarrow x = e^{-s} \\
\int_{0}^{\infty} f(t) e^{-st} dt = F(s)
\\
\text{ https://youtu.be/sZ2qulI6GEk?t=1m5s }
[/math]

🗑️ Anonymous at Wed, 3 Jul 2024 07:40:11 UTC No. 16265619

[math] \displaystyle
\sum_{n=0}^{\infty} a_n x^n = A(x) \\
\sum_{n=0}^{\infty} a(n) x^n = A(x) \\
a(n) \sim \! \sim \! \sim \! \! \! > A(x) \\
a(n) = 1 \sim \! \sim \! \sim \! \! \! > \frac{1}{1-x} \: \: \, ,\: \: \: \left | x \right | < 1 \\
a(n) = \frac{1}{n!} \sim \! \sim \! \sim \! \! \! > e^x \\
\text{continuous analog:} \\
\int_{0}^{\infty} a(t) x^t dt = A(x) \: \: \, ,\: \: \: 0 < x < 1 \Rightarrow ln(x) < 0 \\
x^t = \left ( e^{ln(x)} \right )^t \: \: \, ,\: \: \: -s = ln(x) \Rightarrow x = e^{-s} \\
\int_{0}^{\infty} f(t) e^{-st} dt = F(s)
\\
\text{ https://youtu.be/sZ2qulI6GEk?t=1m5s }
[/math]

Anonymous at Wed, 3 Jul 2024 07:51:48 UTC No. 16265621

>>16265619
[math] \displaystyle
\sum_{n=0}^{\infty} a_n x^n = A(x) \\
\sum_{n=0}^{\infty} a(n) x^n = A(x) \\
a(n) \sim \! \sim \! \sim \! \! \! > A(x) \\
a(n) = 1 \sim \! \sim \! \sim \! \! \! > \frac{1}{1-x} \: \: \, ,\: \: \: \left | x \right | < 1 \\
a(n) = \frac{1}{n!} \sim \! \sim \! \sim \! \! \! > e^x \\
\text{continuous analog:} \\
\int_{0}^{\infty} a(t) x^t dt = A(x) \: \: \, ,\: \: \: 0 < x < 1 \Rightarrow ln(x) < 0 \\
x^t = \left ( e^{ln(x)} \right )^t \: \: \, ,\: \: \: -s = ln(x) \Rightarrow x = e^{-s} \\
\int_{0}^{\infty} f(t) e^{-st} dt = F(s)[/math]
https://youtu.be/sZ2qulI6GEk?t=1m5s

Anonymous at Wed, 3 Jul 2024 08:08:02 UTC No. 16265637

>>16264744
The definition doesn't involve infinity, you dipshit.

Anonymous at Wed, 3 Jul 2024 08:12:57 UTC No. 16265643

4chan peppers long uninterrupted sequences of characters with <wbr> html tags (this tells the browser that it's fine to line break in the middle of the sequence, which prevents the screen from going wide when people type in AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA) but has the side effect of breaking the Latex parsing.

Anonymous at Wed, 3 Jul 2024 08:17:01 UTC No. 16265649

>>16265643
>uninterrupted equences of characters
simple solution: always put a space before \ or \\

Anonymous at Wed, 3 Jul 2024 08:29:47 UTC No. 16265656

[math] \displaystyle
\boxed{ \mathbb{T} \;
\boxed{ \mathbb{S} \;
\boxed{ \mathbb{O} \;
\boxed{ \mathbb{H} \;
\boxed{ \mathbb{C} \;
\boxed{ \mathbb{R} \;
\boxed{ \mathbb{Q} \;
\boxed{ \mathbb{Z} \;
\boxed{ \mathbb{N}}}}}}}}}}
[/math]

Anonymous at Wed, 3 Jul 2024 09:07:25 UTC No. 16265702

>>16265637
then why is the symbol being used?

Anonymous at Wed, 3 Jul 2024 09:50:07 UTC No. 16265735

>>16265702
"i can see russia from here"
doesn't mean you're in russia yet

Anonymous at Wed, 3 Jul 2024 10:00:49 UTC No. 16265740

>>16265702
You don't necessarily have to refer to infinity in the definition, the symbol is used because that's the notation for a limit of a variable that gets arbitrarily large, in this case the index variable n. Arbitrarily large means exactly that, we just set n to equal larger and larger numbers since we're saying that a certain condition is met for all natural numbers greater than N (this is the off to infinity part).

Anonymous at Wed, 3 Jul 2024 10:25:27 UTC No. 16265770

>>16264395
Filtered.
You will never be Norman because even Norman understands this basic shit within the analytic framework despite his qualms.

Anonymous at Wed, 3 Jul 2024 10:41:41 UTC No. 16265780

I will defend OP. He got filtered because american calculus pedagogy is shit. Most of it is "intuitive" and computation based with a bit of half-baked weakly motivated theory peppered throughout. Kids can pass the entire calculus series without ever learning what a delta-epsilon proof is because they're never forced to. I hope in the future there's a movement to scrap the idea of "Calc 1-3" entirely and and just do proofs + intro to analysis

Anonymous at Wed, 3 Jul 2024 10:54:00 UTC No. 16265794

>>16265780
American Calc I & II: "As you'll learn in more advanced when you take real analysis..."
American real analysis: "As you learned in Calc I & II..."

Even the Stewart-tier books teach d-e proofs within the first few chapters, but yeah, it seems to be a coin toss whether any particular college is going to skip them.

Anonymous at Wed, 3 Jul 2024 14:28:07 UTC No. 16265986

>>16265702
It's just notation.

Anonymous at Wed, 3 Jul 2024 14:35:13 UTC No. 16265994

>>16265656
Obfuscates systems such as the Gaussian integers and complex rationals
Implies that the Cayley-Dickson algebras are related to the constructions of N, Z, Q, and R

Anonymous at Wed, 3 Jul 2024 14:45:00 UTC No. 16266014

>>16265994
It doesn't do either of those things, you idiot.

Anonymous at Wed, 3 Jul 2024 22:18:50 UTC No. 16266760

>>16265986
define the notation.

Anonymous at Wed, 3 Jul 2024 23:16:50 UTC No. 16266844

>>16265616
huh pretty cool

Dunko at Wed, 3 Jul 2024 23:37:19 UTC No. 16266874

>>16266760
The notation's literally defined in the OP pic.

Anonymous at Wed, 3 Jul 2024 23:46:28 UTC No. 16266886

>>16264395
this shit doesn't matter. i can find the limit of a function without understanding what this says. just plug and compute using limit laws

Anonymous at Thu, 4 Jul 2024 00:12:51 UTC No. 16266905

>>16266886
enter non-continuous functions

Anonymous at Thu, 4 Jul 2024 00:55:26 UTC No. 16266958

>>16266905
anytime i see x/0 i can just stop the computation and say that the function is discontinuous. simple as

Anonymous at Thu, 4 Jul 2024 07:01:49 UTC No. 16267331

>>16264395
you aint seen nothing yet

Anonymous at Thu, 4 Jul 2024 14:47:09 UTC No. 16267649

>>16265780
the issue is d-e is pretty intuitive too. hell, i'd argue pretty much all analysis is "intuitive" if you tolerate enough chewing. but with the semester structure, i get why a student wouldn't really care about it if they weren't forced to.

it's that op is now trying to self-study (assuming his semester's over) calculus and is getting filtered by d-e that's stupid.