Analysis

I was a big fan of Series in Calc 2 this past semester, is there any other type of math higher up that is similar to series? I am compelled to learn and do more with Convergence and Divergence.
What books do you suggest?
Pic somewhat but not really related

Other urls found in this thread:

en.wikipedia.org/wiki/Parseval's_theorem
en.wikipedia.org/wiki/Two's_complement
en.wikipedia.org/wiki/Hilbert_space
en.wikipedia.org/wiki/Riemann_sphere
en.wikipedia.org/wiki/Alexandroff_extension
en.wikipedia.org/wiki/Zeta_function_regularization
en.wikipedia.org/wiki/Renormalization
en.wikipedia.org/wiki/P-adic_order#.7F.27.22.60UNIQ--postMath-00000028-QINU.60.22.27.7F-adic_absolute_value
en.wikipedia.org/wiki/Ostrowski's_theorem
twitter.com/SFWRedditImages

analytic number theory

Euclidean elements

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I'm always annoyed when people define the Zeta function like that, it isn't the complete definition of what it is.

>I'm always annoyed when people define the Zeta function like that

...

1+2+3 .. = -1/12 amirite

absolutely

Because I don't have the time to actually add it or see where it is going so you can just say it is anything right?

analysis, as you said. try Tao's Analysis I for a great introduction

Thanks for a legit answer!

Anything Fourier series.
Basically it is an infinite dimensional vector space where the basis vectors are the functions sin's and cos's of all integer frequencies.

en.wikipedia.org/wiki/Parseval's_theorem

Dirichlet kernel and Fejer kernel for convergence criteria

Hell give Euler-Maclaurin formula a try

The subtleties of complex exponents and branch cuts...

Yes

You are ok with 0.33333...=1/3
Are you ok with ...99999.0=-1?

en.wikipedia.org/wiki/Two's_complement

For inspiration

>infinite dimensional vector space
>basis
No such thing.

en.wikipedia.org/wiki/Hilbert_space

SMD

So? That still isn't proof that infinity = -1/12

>SMD
Shaking my dick?

Know the context.
Learn some Complex Analysis

en.wikipedia.org/wiki/Riemann_sphere

en.wikipedia.org/wiki/Alexandroff_extension

Sums get to infinity then wrap around (kind of like how the graph of y=1/x "jumps" from negative infinity to positive infinity)

See My Disproof

>Sums get to infinity then wrap around (kind of like how the graph of y=1/x "jumps" from negative infinity to positive infinity)

I know all that already and none of it justifies saying that infinity is -1/12

Nature is ok with it. I don't think you understand infinity.

en.wikipedia.org/wiki/Zeta_function_regularization

en.wikipedia.org/wiki/Renormalization

>I don't think you understand infinity.
>I don't think
Convincing yourself in your head that 1+2+3... = -1/12 is pretty much the definition of not thinking, wouldn't you say?

There are other absolute values used for convergence criteria that do not resemble your intuitive notion of distance.

en.wikipedia.org/wiki/P-adic_order#.7F.27.22.60UNIQ--postMath-00000028-QINU.60.22.27.7F-adic_absolute_value

en.wikipedia.org/wiki/Ostrowski's_theorem

You're just trying to throw sand in my eyes at this point.

It's already there

You've done nothing but show me things I already know that don't prove infinity = -1/12

"Morally the truth" and "literally the truth" are two separate things, the former only means that you in your mind somehow got it twisted enough to make it seem as though something as absurd as infinity = -1/12 is true. [math]\sum_{x\to\infty}x=\infty[/math] no matter how you want to look at it.

If you really want to keep deluding yourself then go ahead, see how well it works out for you in life.

>If you really want to keep deluding yourself then go ahead, see how well it works out for you in life.
Likewise

>proof that infinity = -1/12
I think you are misunderstanding the idea. The infinite series 1+2+... doesn't equal anything at all. We are assigning values to it in a purely formal way, and there are many ways to assign them. One way is by using the limit of partial sums. For any number we can add up a finite number of terms to exceed it. Another way is by analytic continuation of the zeta function.

Are you retarded? All vector spaces have a basis (Lemma of Zorn or AC).
No infinite dim. Banachspace has a countable basis though.

Idiot. An orthogonal basis is NOT a basis.

>he believes in the axiom of choice
you know how I know you're an undergrad?

Analysis and Combinatorics

Not quite series in the traditional sense but generating functions deal with summations.

Also analysis in general of course.