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Calculatrice de séries de Taylor et de Maclaurin (séries entières)
Calculer la série de Taylor/Maclaurin pas à pas
La calculatrice déterminera le développement en série de Taylor (ou en série entière) de la fonction donnée autour du point donné, avec les étapes affichées. Vous pouvez spécifier l’ordre du polynôme de Taylor. Si vous voulez le polynôme de Maclaurin, réglez simplement le point à $$$0$$$.
Solution
Your input: calculate the Taylor (Maclaurin) series of $$$x \ln{\left(x \right)}$$$ up to $$$n=5$$$
A Maclaurin series is given by $$$f\left(x\right)=\sum\limits_{k=0}^{\infty}\frac{f^{(k)}\left(a\right)}{k!}x^k$$$
In our case, $$$f\left(x\right) \approx P\left(x\right) = \sum\limits_{k=0}^{n}\frac{f^{(k)}\left(a\right)}{k!}x^k=\sum\limits_{k=0}^{5}\frac{f^{(k)}\left(a\right)}{k!}x^k$$$
So, what we need to do to get the desired polynomial is to calculate the derivatives, evaluate them at the given point, and plug the results into the given formula.
$$$f^{(0)}\left(x\right)=f\left(x\right)=x \ln{\left(x \right)}$$$
Evaluate the function at the point: $$$f\left(0\right)=0$$$
Find the 1st derivative: $$$f^{(1)}\left(x\right)=\left(f^{(0)}\left(x\right)\right)^{\prime}=\left(x \ln{\left(x \right)}\right)^{\prime}=\ln{\left(x \right)} + 1$$$ (steps can be seen here).
Evaluate the 1st derivative at the given point: as can be seen, the 1st derivative does not exist at the given point.
Answer: the Taylor (Maclaurin) series can't be found at the given point.