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1) Sign up with your mobile and enter the code «coins». Click on «claim», wait for the verification.
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Our website uses cookies to offer you a better browsing experience. Find out more on how we use cookies and how you can control them by clicking here. By continuing to use this site, you are agreeing to the use of cookies.OKQ:
Equivalent definitions of Taylor series
If the Taylor series of $f$ centered at $x_0$ is defined as:
$f(x)=\sum_{n=0}^{+\infty} \frac{1}{n!} (x-x_0)^n \cdot f^{(n)}(x_0)$,
for any $x \in \mathbb{R}$,
what is the equivalence between the following definitions of the Taylor series of $f$ at $x_0$:
The one defined above in 1)
The one defined as $\sum_{n=0}^{+\infty} \frac{1}{n!} (x-x_0)^n \cdot f^{(n)}(x_0)$
Thanks!
A:
I don’t know about this definition but I have heard about this notation. If the derivative at $x_0$ exists and is continuous around the point you are considering. Then using the existance of derivatives, you can write
$$f^{(n)}(x_0)=(-1)^n\binom{n}{0}(x-x_0)^n+(-1)^{n-1}\binom{n}{1}(x-x_0)^{n-1}+\cdots+(-1)^1\binom{n}{n-1}(
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published: 18 Apr 2017
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