Sequences and Series

We typically denote sequences and series with a name like this: $${a_n}=\{1, 2, 4, 8,...\}$$ Most of the time, we can also write them as functions: $$a_n=f(n).$$ For example, $a_n=\{3^n\}$. We can define sequences recursively: $a_n=f(a_{n-1})$. (note: "$n$" is typically used for counting numbers (integers). So, $f(n)$ is typically discrete, whereas $f(x)$ is typically a continuous function).

A series converges if $\lim_{n\to\infty}S_n=L\in \mathbb{R}$. Otherwise, it diverges.

Geometric Series
A geometric series is a series of a sequence of a constant times a ratio to the $n$th: $$\sum_{n=1}^\infty ar^{n-1}.$$ If we consider $s_n=a+ar+ar^2+\dots+ar^{n-1}$, then we'll see: $$s_n(1-r)=a-ar^n,$$ which (when $r\neq1$) means $$s_n=\frac{a-ar^n}{1-r}.$$ To determine whether the geometric series diverges, we need to determine whether the limit of $s_n$ as $n$ approaches infinity is a number. Think about it: if $|r|>1$, then the numerator will approach infinity (or negative infinity). So, a geometric series converges if and only if $|r|< 1$. In that case, the limit is simply $\frac{a}{1-r}$! (p.s. obviously, if $r=1$, we just take the infinite sum: $s_n=an$ as $n$ approaches infinity... which clearly diverges.)

(check out this video if you want a person to explain geometric series)

This test for geometric series (known as the geometric series test) is one of a number of convergence tests.

Convergence Tests

1. Divergence Test. If $\lim_{n\to\infty}a_n\neq 0$, then $\sum a_n$ diverges. Use this test before all others! However, remember that the Divergence Test is not an "if and only if" statement — the only thing it can guarantee is whether a given series diverges (just because the test doesn't say it diverges doesn't mean it doesn't — consider the harmonic series).
2. Geometric Series. A geometric series $\sum ar^{n-1}$ converges if and only if $|r|< 1$.
3. Comparison Test. Suppose that $0\leq a_n\leq b_n$ and $\sum b_n$ converges. Then, $\sum a_n$ converges. By similar logic, if $a_n\geq c_n$ and $\sum c_n$ diverges, then $\sum a_n$ also diverges.
4. Integral Test. Suppose $f(x)$ is a continuous, positive, decreasing (all three of these must hold!) function for $x \geq 1$, and that $f(n)$ represents the terms of the sequence $a_n$ for which you are summing the series $\sum a_n$. Then, the series $\sum a_n$ converges if and only if the improper integral $\int_1^\infty f(x)dx$ converges. I recommend watching Dr. Trefor Bazett's video about this test for a visual representation.
5. Limit Comparison Test. If the ratio $\frac{a_n}{b_n}$ approaches a positive limit $L$, then $\sum a_n$ and $\sum b_n$ both either converge or both diverge.
6. Ratio Test. If we have the series $\sum a_n$, we can decide whether it converges by considering the following: $L=\lim_{n\to\infty} |\frac{a_{n+1}}{a_n}|$. If $L< 1$, the series absolutely converges; if $L> 1$, it diverges; and if $L = 1$, the series may be divergent, conditionally convergent, or absolutely convergent.
7. Root Test. Same as the Ratio Test, but $L=\lim_{n\to\infty}\left | a_n\right |^{\frac1n}.$
8. Alternating Series Test. An alternating series is one where the terms can be written as $a_n=(-1)^n b_n$ or $a_n=(-1)^{n + 1} b_n.$ $a_n$ converges if $\lim_{n\to\infty} a_n=0$ and $\{b_n\}$ is a decreasing sequence.

Taylor Series

How do we actually calculate $\cos(x)$ for all $x$? How do we know that $\cos(39^\circ)\approx 0.777$? In computers, we obviously have a few optimizations (say, when $x\approx 0^\circ,45^\circ,$ or $60^\circ$). However, we don't really know how to find the exact value of a function like $\cos$ that isn't easily representable. This is where we decided to use the magic of approximation. If a function $f(x)$ is continuous and infinitely differentiable (and we can find its derivatives), we can model it with something called a Taylor series that gives us a polynomial approximation of the function.

Taylor Series
For an infinitely differentiable function $f$, its Taylor polynomial centered at $x=a$ is $$\begin{align*}f(x)&=\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n\\&=f(a)+f'(a)(x-a)+\frac{f''(a)}{2}(x-a)^2+\frac{f'''(a)}{6}(x-a)^3+\dots\end{align*}$$ (See Paul's Online Math Notes for an explanation of how this is found.)

When $a=0,$ the series is called the Maclaurin series.

Then, we can also find the error in Taylor approximation: $$R_n(x)=f(x)-T_n(x),$$ Which means that we can write $$f(x)=T_n(x)+R_n(x).$$

Ex. Find the Taylor series for $e^x$ around $x=0.$ (then Google the answer, I ain't typing allat out.)