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Rekenmachine voor Taylor- en Maclaurin (macht)reeksen
Bepaal de Taylor/Maclaurin-reeks stap voor stap
De rekenmachine bepaalt de Taylorreeks (of machtreeks) van de gegeven functie rond het gegeven punt, waarbij de stappen worden getoond. Je kunt de orde van het Taylorpolynoom opgeven. Als je het Maclaurinpolynoom wilt, stel het punt dan in op $$$0$$$.
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.