How do you demonstrate each $o(x^n)$ term on Taylor Series?

Asked Feb 20 '24 at 14:55

Active Feb 20 '24 at 15:04

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I found how to demonstrate the Taylor Series with integral remainder.

$$ f(x) = \sum_{k=0}^n \Biggl[\frac{f^{(k)}(a)}{k!}(x-a)^k \Biggr] + \int_a^x \frac{f^{(n +1)}(t)}{n!}(x-t)^n dt $$

But I do you pass to this formula to the formula with the $o(x^n)$ term ?

$$ f(x) = \sum_{k=0}^n \Biggl[\frac{f^{(k)}(a)}{k!}(x-a)^k \Biggr] + o_a((x-a)^n) $$

How is it possible to demonstrate it ?

Thank you for your help.

edited Feb 20 '24 at 15:04

J.G.

asked Feb 20 '24 at 14:55

jozinho22

Under the hypothesis for the integral remainder ($f$ is $\mathcal C^{n+1}$ on an interval around $a$), use that $f^{(n+1)}$ is bounded on any smaller closed interval: your rest becomes more precisely $O_a((x-a)^{n+1})$. Note however that the "formula with the $o(x^n)$ term" can be proved directly under the much weaker hypothesis of existence of $f^{(n)}(a)$. – Anne Bauval Feb 20 '24 at 15:04
Prove the Lagrange form. – J.G. Feb 20 '24 at 15:06

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