An infinite series of polynomials representing a function, generally expressed as $\displaystyle\sum_{n=0}^{\infty}{a_n(x-r)}$.

$a_n$ is a sequence of real numbers in the power series $\displaystyle\sum_{n=0}^{\infty}{a_n(x-r)}$.

'r' represents a real number in the power series $\displaystyle\sum_{n=0}^{\infty}{a_n(x-r)}$.

Multiply the power series of $e^x$ by $x^2$.

Take the derivative of each term in the power series of $f(x)$.

$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots+\frac{x^n}{n!}$

$\cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \ldots+\frac{(-1)^nx^{2n}}{(2n)!}$

$\sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots+\frac{(-1)^nx^{2n+1}}{(2n+1)!}$