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泰勒和麦克劳林（幂）级数计算器
逐步求泰勒/麦克劳林级数
该计算器将求出所给函数在给定点处的泰勒（或幂）级数展开，并显示步骤。你可以指定泰勒多项式的阶数。如果需要麦克劳林多项式，只需将点设为 $$$0$$$。
Solution
Your input: calculate the Taylor (Maclaurin) series of $$$\frac{1}{x}$$$ 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.
$$$f^{(0)}\left(x\right)=f\left(x\right)=\frac{1}{x}$$$
Evaluate the function at the point: as can be seen, the function does not exist at the given point.
Answer: the Taylor (Maclaurin) series can't be found at the given point.