What's 1+1? (1 Viewer)

Queenroot · Oct 28, 2015

sadpwner said:
I wouldn't know because I don't do math anymore nor school? Time to slowly forget everything I've learned.

what are you going to do in uni?

Crisium · Oct 28, 2015

Drsoccerball said:
lol why do you know that?

It's InteGrand he knows everything

Zen2613 · Oct 28, 2015

InteGrand said:
$We want to say a function $f(x)$ can be written as $f(x) = c_0 + c_1 x + c_2 x^2 + \cdots $ (Taylor series centred about 0, i.e. Maclaurin series, where $|x|<R$, where $R$ is the radius of convergence). If it can be written this way, then differentiating $k$ times we have$

$$f^{(k)}(x) = 0 + 0 + \cdots + 0 + k! c_k+ c_{k+1} (k+1)k(k-1)\ldots 2x^{k+1}+\text{etc}$, where the etc. means further higher order terms in $x$, which will vanish if we put $x=0$. So putting $x=0$ gives $f^{(k)}(0) = k! c_k + 0 + 0 +\cdots \Rightarrow c_k = \frac{f^{(k)}}{k!}$, for $k=0,1,2,\ldots$, where $f^{(k)}(x)$ is the $k$th derivative of $f$ evaluated at $x$.$

$This doesn't necessarily mean that $f$ \textit{can} be written this way, it just means that \textit{if} it can, then the coefficients are as given above ($k$th derivative at $x$ divided by $k$ factorial). There are other theorems regarding whether the function can be written that way.$

How do you know whether a function can be written like that ? Actually wait the explanation is probably far too deep and complex for this blog, what are the theorems called so I can look them up ?

Drsoccerball · Oct 28, 2015

Zen2613 said:
How do you know whether a function can be written like that ? Actually wait the explanation is probably far too deep and complex for this blog, what are the theorems called so I can look them up ?

It's called a Taylor Series.

Zen2613 · Oct 28, 2015

Drsoccerball said:
It's called a Taylor Series.

Yeah i know (a little) about taylor series, but I've never understood why you can let a function be a polynomial be a polynomial of infinite degree in the first place. Like say to find the e^x one, let e^x = a0 +a1x +a2x^2 +... And then use calculus to find the coefficents. But how do you know that e^x can be written in this form in the first place ? We need some theorems!

MathsN00b · Oct 28, 2015

1 + 1 = a retarded window
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MathsN00b · Oct 28, 2015

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InteGrand · Oct 28, 2015

Zen2613 said:
Yeah i know (a little) about taylor series, but I've never understood why you can let a function be a polynomial be a polynomial of infinite degree in the first place. Like say to find the e^x one, let e^x = a0 +a1x +a2x^2 +... And then use calculus to find the coefficents. But how do you know that e^x can be written in this form in the first place ? We need some theorems!

Search up Taylor's theorem.

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What's 1+1? (1 Viewer)

Queenroot

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