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Taylor and Maclaurin (Power) Series Calculator
Find Taylor/Maclaurin series step by step
The calculator will find the Taylor (or power) series expansion of the given function around the given point, with steps shown. You can specify the order of the Taylor polynomial. If you want the Maclaurin polynomial, just set the point to $$$0$$$.
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.