Common Series:

Geometric Series:

A geometric series is defined by the following:$$\sum\limits_{n=0}^{\infty}ar^n$$

If $a \neq 0$ and $| r | < 1$ , then the geometric series converges to:

\sum_{n = 0}^{\infty} a r^{n} = \frac{a}{1 - r}

where $a$ is equal to the first term of the sequence.

If $| r | \geq 1$ the series diverges.

P-Series:

A p-series is defined by the following:

\sum\limits_{n=0}^{\infty}\frac{1}{n^p}$$where $p$ is any real constant.  If $p>1$ the series *converges*. If $p\le1$ the series *diverges*   # Convergence Tests:  Follow this order to help determine what test to use:  1.Nth Term Divergence Test: > This checks for general divergence, if the series does in fact diverge then no further work is needed.   2.Ratio Test&Root Test> This is meant to deal with both positive and negative series, if for some reason they do not work or don't seem applicable move on to the next text  3.Comparison Tests> This is mostly helpful when you know of a larger p-series or geometric series that is similar to the series and it converges.  4.Integral Test> Try to avoid using this as it involves many assumptions but if you see a series that meets the criteria and is easily integrate able you can use this  5.Absolute Convergence&Alternating Series Test(For Negative Series) > Use these sparingly  # Power Series  A power series is an infinite series has a center $c$, radius of convergence $R$, and interval of convergence $I$.  The *radius* is defined by   $$\left|x-c\right|<R

The interval of convergence is defined by

(c - R, c + R) or [c - R, c + R] or (c - R, c + R] or [c - R, c + R)

This is because the series may or may not converge at the endpoints of the interval.

Taylor Polynomials:

The idea behind a Taylor Polynomial is to extend the idea of linearization. This allows for a function to be approximated in the form of a polynomial, which is much easier to calculate and derive, and integrate.

A Taylor Polynomial is a partial sum of a Taylor Series that approximates the values of a function centered at a point $c$ up to a value $n$ .

T_{n} (x) = \sum_{i = 0}^{n} \frac{f^{(i)} (c)}{i!} (x - c)^{i}

This can also be represented as:

T_{n} (x) = f (c) + \frac{f^{'} (c)}{1!} (x - c) + \frac{f^{″} (c)}{2!} (x - c)^{2} + \frac{f^{‴} (c)}{3!} (x - c)^{3} . . . \frac{f^{n} (c)}{n!} (x - c)^{n}

Maclaurin Polynomials:

This is just a Taylor Polynomial but the center $c$ is assumed to be 0.

Taylor Series:

A Taylor Series is an infinite sum of a sequence that allows for the values of a function around a point by extending the idea of linearization around a center $c$ .

It is given by the following:

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

Finding the Taylor Series representation of a function is the same as finding the power series representation of a function.

Maclaurin Series:

This is just a Taylor Series but the center $c$ is assumed to be 0.

Common Maclaurin Series:

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

$R = \infty I = (- \infty, \infty)$

\arctan (x) = \sum_{n = 0}^{\infty} (- 1)^{n} \frac{x^{2 n + 1}}{2 n + 1}

$R = 1 I = (- 1, 1)$

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

$R = 1 I = (- 1, 1)$

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

$R = \infty I = (- \infty, \infty)$

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

$R = \infty I = (- \infty, \infty)$