Question

How do you find the power series for a function centred at $c$ ?

Answer 1

We have a direct formula for finding the power series for a function centred at $c$. A power series is an infinite series of any form. Power series are useful in mathematical analysis, where they arise in the form of Taylor series of Infinitely differentiable functions altogether. Also, power series of any function are Taylor series of some smooth functions.

Complete step by step answer:
A function can be easily expressed as a power series around any center c, although all but finitely many of the coefficients will be zero since a power series has infinitely many defining terminologies.Taylor collection of a characteristic is an endless sum of phrases which are expressed in phrases of the characteristic's derivatives at an unmarried point. For maximum not unusual place functions, the characteristic and the sum of its Taylor collection are the same close to this point.

The Taylor series for the given function which is catered at $c$ is
$f(x) = \sum\limits_{n = 0}^\infty {\dfrac{{{f^{(n)}}(c)}}{{n!}}} {(x - c)^n}$.
Where $f\left( x \right)$ is the given function.$C$ is the constant, where the function is centered and $n$ is the number of terms expected in the power series. This is how we find the power series for a given function who has center at c.

Note: When the given function is centered at zero. Then, the power series is generated through the Maclaurin series. When expanding the series, we have to be careful while assigning the value of the power as they increase gradually.