Sequences and Series

Arithmetic Sequences

An arithmetic sequence is a sequence where the common difference (d) between consecutive terms is constant.

d=T_{2}-T_{1}=T_{3}-T_{2}=T_{n}-T_{n-1}

The general term of an arithmetic sequence is

T_{n} = a+(n-1)d

where a is the first term and d is the common difference.

Geometric Sequences

A geometric sequence is a sequence where the common ratio (r) between consecutive terms is constant.

r= \frac{T_{2}}{T_{1}}=\frac{T_{3}}{T_{2}}=\frac{T_{n}}{T_{n-1}}

The general term of a geometric sequence is

T_{n} = ar^{n-1}

where a is the first term and r is the common ratio.

Quadratic Sequences

A quadratic sequence is a sequence where the the second difference is common. Below is an example.

The general term of a quadratic sequence is

T_{n} = an^{2}+bn+c

then 2a = second difference.

3a+b = T_{2}-T_{1}

a+b+c = first term.

Taylor Series

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

Maclaurin Series

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