# Precalculus

Rules of Thumb

Sequences are ordered lists, series are sums of an ordered list
Two types of each: Arithmetic (add/subtract, uses a common difference) and Geometric (multiply/divide/raise to a power, uses a common ratio)
For more basic rules as well as some formulas of sequences and series, refer back to the Algebra II Review on the subject
Remember: $e$ can be expressed as an infinite series:$\displaystyle\sum_{n=0}^\infty \frac{1}{n!}$

What to Review

Sigma notation
Factorials

Infinite Series

These are series that go infinitely onward, expressed by $\displaystyle\sum_{i=a}^\infty$. Surprisingly enough, these series do sometimes have a finite sum.
This sum can be found by the formula $\frac{a_1}{1-q}$, but only as long as it satisfies the condition of the common ratio being $-1< q< 1$ (converges).

Convergent vs. Divergent Series

Convergent- the series tends toward an “asymptote”. The more numbers you add the closer the output gets to that limit. For this to happen, the common ratio must be between -1 and 1 (a positive or negative fraction).
An example is the instance $\displaystyle\sum_{i=0}^\infty a_n=\frac{(-1)^{n+1}}{n}$ (if you want to see if this works, remember PEMDAS)
If it doesn’t approach a limit, or approaches infinity, then the series diverges

Binomial Expansion

The Binomial Theorem: this is a surefire, albeit very tedious process of expanding binomials to any power you like. However, I would not recommend using it unless you have a huge power such as 100 because of time constraints and the probability of error. Here it is:
$(a+b)^n=\displaystyle\sum_{i=0}^n \binom{n}{i} a^{n-i}b^i$ where $\displaystyle\binom{n}{i}=C_i^n=\frac{n!}{(n-i)!i!}$ ( $C_i^n$ means $n$ choose $i$)
For easier and quicker expansion, use Pascal’s Triangle, which can also be found in the Algebra II review. The pattern for the triangle is: starting with one add the number around it and write them beneath it (blank spaces count as 0)