泰勒和麦克劳林（幂）级数计算器

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: $$$f\left(0\right)=0$$$

1. 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.