What is it?

[plush arch]

How use?

[wooden apex]

1. Ask your question and show the work you've done so far. If you've posted a screenshot of a question, specify which part you need help with.
2. Wait patiently for a helper to come along.
3. Once someone helps you, say thank you and close the thread with:

   +close
   

4. Feel free to nominate the person for helper of the week in #helper-nominations
5. Do not ping the mods, unless someone is breaking the rules.
6. If you're happy with the help you got here, and the server overall, you can contribute financially as well:

[plush arch]

[cedar basin]

...where did this even come from?

[molten nebula]

$a_n \sim b_n$ when both are nonzero and $\frac{a_n}{b_n} \xrightarrow[n \to \infty]{} 1$

[wispy hatch]

Rion

[molten nebula]

In nonrigorous terms, they behave roughly the same when n tends to infinity

[plush arch]

example?

[molten nebula]

$a_n = n^2$, $b_n = n^2 + n + 1$

[wispy hatch]

Rion

[plush arch]

Oh

[molten nebula]

$a_n = \sum_{k=1}^{n} \frac{1}{k}, b_n = \ln n$

[wispy hatch]

Rion

[plush arch]

OHHHHHHH

[molten nebula]

$a_n = n!, b_n = \sqrt{2\pi n} \left( \frac{n}{e}\right)^n$

[wispy hatch]

Rion

[molten nebula]

last one is a tough one to prove

[plush arch]

[molten nebula]

You can look up proofs on the internet because it's too long for me to write it here

should you need one

[plush arch]

Thanks 🥲👍

[coral bolt]

@plush arch has given 1 rep to @molten nebula

[dim stag]

f(x) ~ g(x), x -> x0 means that f(x)/g(x) -> 1 when x -> x0.

[plush arch]

what?

[dim stag]

What?

[plush arch]

I don't understand

Help me please 🫤

[dim stag]

Which part?

[plush arch]

1 part

[dim stag]

Ok, which one?

[plush arch]

f(x) ~ g(x)

[plush arch]

g(x)

[dim stag]

f(x) and g(x) are just functions.

[plush arch]

What is the difference?

[dim stag]

What difference?

[plush arch]

f(x) and g(x)

[dim stag]

We're not talking about their difference.

[plush arch]

What is it g(x)?

[dim stag]

As I said, f(x) and g(x) are arbitrary functions.

Why focus on them? Your question was about the Landau symbol ~.

[plush arch]

Oh

Sorry

[plush arch]

What is it ~

[dim stag]

It's one of the Landau symbols. As I said above, f(x) ~ g(x), x -> x0 means that f(x)/g(x) -> 1 when x -> x0.

[plush arch]

Um

[plush arch]

X -> x0?

[dim stag]

The point at which we find the limit also doesn't matter in general. Could be a finite number, could be +∞ (like in your case) or -∞.

[plush arch]

What?

[dim stag]

I don't know what you're asking.

[plush arch]

I hope I'm not a troll, because this is actually the first time I've heard this.

[plush arch]

I don't understand 😕

[dim stag]

I recommend reading some material on Landau notation if you're not familiar with it.

[molten nebula]

It's basically the same thing as I explained earlier, but instead of n going to infinity, we have x going to a certain point

(a point which can also be at infinity, but not necessarily)

for example, consider: (x - 1) and (x² - 1) as x goes to 1

they are not equivalent

[plush arch]

Why?

[cedar basin]

... for the reasons that have been explained in explicit detail in this thread.

[plush arch]

?

[cedar basin]

How am I supposed to answer a question that literally contains no words?

[plush arch]

I think I asked the right question.

[cedar basin]

"The right question" is "?"?

[plush arch]

This means that I did not understand what you wrote

[cedar basin]

Yes, but you haven't told me what you didn't understand about what I wrote.

[plush arch]

All

[cedar basin]

Okay, so the definition of the word "for" is...

Actually, wait, I can't explain this if you don't know how to read.

Or parse English.

[plush arch]

And you don't know how to convey your idea correctly

[cedar basin]

At least I try harder than one word at a time.

You literally said that you didn't understand anything about what I wrote. What am I supposed to do with that?

[plush arch]

^

[cedar basin]

You're giving me zero guidance on how to communicate with you effectively.

[plush arch]

normal explanation.

[cedar basin]

So an explanation at a right angle to a vector?

[plush arch]

Angle?

You write nonsense

[cedar basin]

No, that's the definition of the word "normal" in math. Look it up.

https://en.wikipedia.org/wiki/Normal_(geometry)

[plush arch]

Why do I need it?

[molten nebula]

you can see that the ratio between them doesn't go to 1

$\frac{x^2 - 1}{x-1} = \frac{(x-1)(x+1)}{x-1} = x+1 \xrightarrow[x \to 1]{} 2$

[wispy hatch]

Rion

[plush arch]

X-1
X=2
2-1=1

[molten nebula]

wdym x = 2?

here x is tending to 1

[plush arch]

X=0.9?

[verbal pond]

💀

[delicate talon]

tending is when x goes towards to 1 more an more
like x = 0.99999999999999999999 is tending to 1

[dim stag]

A number isn't tending to another. A sequence or a function can.

[delicate talon]

a variable can tend to a number too cant it? here x tends to 1

[dim stag]

True. Not a constant, though.

[plush arch]

Oh okay

[plush arch]

Thanks 😊