/sci/ - Science & Math

[Anonymous No.8664355]

do series and functions have an interconnected relationship?

[Anonymous No.8664360]

a function is a set
a series is a sum

you can turn some series into functions by singling out a variable, i.e. [math]\sum_{n\geq 0} n^{-2}= \frac{\pi^2}{6} [/math] is just a series but [math] \zeta(s)=\sum_{n\geq 0} n^{-s} [/math] is the Riemann zeta function

[Anonymous No.8664362]

>>8664360
shoulda been [math] n\geq 1 [/math] obviously

[Anonymous No.8664363]

>>8664360
so a series is simply a number where the approach to summation is the defining quality? and a function is simply an algebraic transformation that can take any value? i feel like such a fool thinking there was a deeper connection simply because of the summation.

[Anonymous No.8664370]

>>8664363
>so a series is simply a number where the approach to summation is the defining quality?
yes

https://en.wikipedia.org/wiki/Series_(mathematics)
>For any sequence { a n } {\displaystyle \{a_{n}\}} \{a_{n}\} of rational numbers, real numbers, complex numbers, functions thereof, etc., the associated series is defined as the ordered formal sum...

> and a function is simply an algebraic transformation that can take any value?
no a function from sets [math] X [/math] to [math] Y[/math] is a set of ordered pairs [math] S=\{(a,b) \mid a\in X, b\in Y\} [/math] where [math] (a,b)\in S [/math] and [math] (a,c)\in S[/math] implies [math] b=c [/math] (one output for every input)

there doesn't have to be any obvious algebraic transformation to it, something like [math] \{(0,1), (\pi, 10)\}[/math] is a function too

[Anonymous No.8664378]

>>8664370
would it be unreasonable to propose a series is a set due to the summation being an expression of what would be elements in said set?

[Anonymous No.8664382]

Given a seqence [math] (a_n)_n [/math], then

[math] m \mapsto \sum_{k=1}^m a_n [/math]

is also a function.

And given an indexing [math] a [/math] of element of a monoid M (a set where a binary operation + is defined), i.e. a function
[math] a : {\mathbb M} \to X [/math]
you can define another function
[math] m \mapsto \sum_{k=1}^m a(n) [/math]

I'd advise against the claim of the guy above, defining a function as a set of pairs (set mind detected), but that's in any case a good (albeit raw) model of a function.

[Anonymous No.8664383]

>>8664378
Yes, you could have for instance the partial sums of [math]\sum_{n \geq 1} 2^{-n} [/math] all in a set: [math]\{ 2^{-n} | n \geq 1 \} [/math], and then to find the actual limiting sum, you apply the limit of it, which can be thought of a function from sets to real numbers, and the number you get when you input that set of partial sums, is the liming sum.

This is all close to what are called Cauchy Sequences: https://en.wikipedia.org/wiki/Cauchy_sequence

[Anonymous No.8664385]

>>8664382
what is that m notation? is that's what's called a morphism?

[Anonymous No.8664386]

>>8664378
i guess you could turn a series into a set like this, using [math] \sum{n \geq 1} a_n [/math] to get the set [math] \{ (n,a_n) \mid n\geq 1 \} [math]

the essential difference i guess is that the series denotes a specific number (the sum, if it converges) while a set

>>8664383
this isn't good enough because the sum depends on the order of the sequence (https://en.wikipedia.org/wiki/Riemann_series_theorem))

[Anonymous No.8664387]

Given a seqence [math] (a_n)_n [/math] of elements of a monoid M (a set "M" where a binary operation "+" is defined), then

[math] N \mapsto \sum_{k=1}^N a_n [/math]

is also a function in [math] {\mathbb N} \to M [/math]

A sequence is none other than an indexing of element of M , i.e. a function
[math] a : {\mathbb N} \to M [/math]

PS I'd advise against the claim of the guy above, defining a function as a set of pairs (set mind detected), but that's in any case a good (albeit raw) model of a function.

[Anonymous No.8664388]

>>8664378
i guess you could turn a series into a set like this, using [math] \sum_{n\geq 1} a_n [/math] to get the set [math] \{ (n,a_n) \mid n\geq 1 \} [\math]

the essential difference i guess is that the series denotes a specific number (the sum, if it converges) while a set

>>8664383
this isn't good enough because the sum depends on the order of the sequence (https://en.wikipedia.org/wiki/Riemann_series_theorem))

[Anonymous No.8664390]

>>8664386
>>8664383
so is a sequence different from a series due to the necessity of order? man i feel like there's so much i don't know but want to know.

[Anonymous No.8664391]

>>8664388
>[math] \{ (n,a_n) \mid n\geq 1 \} [/math]

[Anonymous No.8664393]

>>8664390
>so is a sequence different from a series due to the necessity of order?
a sequence and series are both ordered, the difference is whether you're summing up that sequence or not, if you are then you're looking at the series defined by that sequence

[Anonymous No.8664410]

>>8664387
is a a fixed value in this context (1+1+1+1)?

[Anonymous No.8664458]

>>8664390
a sequence is a function, whose domain is the natural numbers
A series is a sequence of partial sums of a sequence <a>
[math] s_{i+1} = a_{i+1} + s_{i} [/math], where [math] s_0 = 0 [/math]