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georg gill
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Does anyone know a proof of taylor formula (actually I am looking for proof for maclaurin series but guess it is the same) without using derivation rules for polynomials?
georg gill said:without using derivation rules for polynomials?
georg gill said:I believe this explains taylor theorem
http://bildr.no/view/1030479
but it uses the rule I mentioned above
This one is the closest proof I have found but it uses the fact alsoStephen Tashi said:I don't know of any proof where that fact doesn't play some role. I glanced at a proof in a book that began by using an integration by parts, but that required knowing how to integrate [itex] (x-t)^n [/itex], which involves knowledge of how to differentiate.
georg gill said:I guess the same problem goes for binomial theorem?
Stephen Tashi said:You should polish your writing skills, georg! Are you asking whether there is a proof of Taylor's formula that does not rely on binomial theorem? Or are you asking whether there is a proof of the binomial theorm that does not rely on the differentiation rules for polynomials? I think the binomial theorem can be proven without using derivatives.
georg gill said:I don't know where I am not being clear :(
georg gill said:Ok I will try to explain it as clear as possible:)
Purpose:
Prove that
[tex](e^x)^y=e^{xy}[/tex]
Stephen Tashi said:As I understand your goal, you wish to prove:
Theorem 1: If [itex] x [/itex] is a real number and [itex] r [/itex] is a rational number then [itex](e^x)^r = e^{xr} [/itex].
Is that correct? Or do you want [itex] x [/itex] to also be a rational number?
georg gill said:So it is the problem with how to differentiate polynomials in link two (II) that I can't prove here
dimension10 said:But I had to use the elementary power rule. But why do you not want to use it? You can easily prove it.
[tex]y=x^n[/tex]
[tex]\mbox{ln}y=n\mbox{ln}x[/tex]
[tex]\frac{1}{y}\frac{\mbox{d}y}{\mbox{d}x}=\frac{n}{x}[/tex]
[tex]\frac{\mbox{d}y}{\mbox{d}x}=\frac{nx^n}{n}[/tex]
[tex]\frac{\mbox{d}y}{\mbox{d}x}=n{x}^{n-1}[/tex]
georg gill said:You use the rule for log with any base:
[tex]log_x a^x=x log_x a[/tex]
I wanted to prove 4 in link here
http://bildr.no/view/1031000 (t)
there i used the log rule you used so I had to prove that one as well. I tried proving it with derivation:
http://bildr.no/view/1031585
but it relies on among others rules rule for differntiation of polynomials. The other rules I can prove but the proof for power rule relies on log rule and then I can't prove the log rule this way.
I have tried to explain it more clear in post number 12 on the first page of this thread
Someone said earlier in this thread that it was a proof for 4 (4 is in the first link (t) in this post) that used among other things dedekinds cut to prove it for all real numbers if someone know where I could find it or buy it online I would be very thankful!
dimension10 said:[tex]\mbox{We know that } e=\lim_{n\rightarrow\infty}{\left(1+\frac{1}{n} \right)}
^{n}=\lim_{h \rightarrow 0}\left(1+h\right)^\frac{1}{h}[/tex]
georg gill said:I need to prove lhopitals I am working on it
The Proof Taylor Formula is a mathematical theorem that allows for the approximation of functions using polynomials. It is commonly used in calculus and analysis to find the value of a function at a specific point.
The Proof Taylor Formula has many applications in mathematics, physics, and engineering. It can be used to solve differential equations, find maximum and minimum values of functions, and calculate derivatives and integrals.
The Proof Taylor Formula can be derived using the concept of infinite series. It involves taking the derivatives of a function at a specific point and using these values to create a polynomial approximation of the function.
No, the Proof Taylor Formula can only be used for functions that are infinitely differentiable, meaning that they have derivatives of all orders at all points. Functions with discontinuities or sharp corners cannot be approximated using this formula.
No, the Proof Taylor Formula can only provide an approximation of a function at a specific point. As the number of terms in the polynomial increases, the approximation becomes more accurate, but it will never be an exact value.