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Introduction to Power Series

Key Questions

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

Taylor Series centered at c#

$f (x) = \infty \sum n = 0 \frac{f^{(n)} (c)}{n!} {(x - c)}^{n}$ $f(x)=sum_{n=0}^infty {f^{(n)}(c)}/{n!}(x-c)^n$

I hope that this was helpful.

Wataru · · Oct 10 2014
What are the most important power series to memorise?

Useful Maclaurin Series

$\frac{1}{1 - x} = \infty \sum n = 0 x^{n}$ $1/{1-x}=sum_{n=0}^infty x^n$

$e^{x} = \infty \sum n = 0 \frac{x^{n}}{n!}$ $e^x=sum_{n=0}^infty{x^n}/{n!}$

$sin x = \infty \sum n = 0 {(- 1)}^{n} \frac{x^{2 n + 1}}{(2 n + 1)!}$ $sinx=sum_{n=0}^infty(-1)^n{x^{2n+1}}/{(2n+1)!}$

$cos x = \infty \sum n = 0 {(- 1)}^{n} \frac{x^{2 n}}{(2 n)!}$ $cosx=sum_{n=0}^infty(-1)^n{x^{2n}}/{(2n)!}$

I hope that this was helpful.

Wataru · · Oct 19 2014
What is a Power Series?

You can thing of a power series as a polynomial function of infinite degree since it looks like this:

$\infty \sum n = 0 a_{n} x^{n} = a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + \dots$ $sum_{n=0}^inftya_nx^n=a_0+a_1x+a_2x^2+a_3x^3+cdots$

I hope that this was sufficient.

Wataru · · Sep 26 2014

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Power Series

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