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Calculadora de series de Taylor y de Maclaurin (de potencias)
Calcular la serie de Taylor/Maclaurin paso a paso
La calculadora encontrará el desarrollo en serie de Taylor (o serie de potencias) de la función dada alrededor del punto dado, mostrando los pasos. Puede especificar el orden del polinomio de Taylor. Si desea el polinomio de Maclaurin, simplemente establezca el punto en $$$0$$$.
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.