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

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

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

1. Start with $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$. 2. Multiply each term by $x^2$. 3. Result: $x^2 + x^3 + \frac{x^4}{2!} + \frac{x^5}{3!} + ... + \frac{x^{n+2}}{n!} + ...$

1. Start with $h(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + ... + \frac{(-1)^nx^{2n}}{(2n)!} + ...$. 2. Take the derivative of each term. 3. Result: $-x + \frac{x^3}{3!} - \frac{x^5}{5!} + ... + \frac{(-1)^nx^{2n-1}}{(2n-1)!} + ...$

1. Take the derivative of each term in $f(x)$. 2. $f'(x) = 1 - x^2 + x^4 - x^6 + ... + (-1)^nx^{2n} + ...$

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 $\displaystyle\sum_{n=0}^{\infty}{a_n(x-r)}$.