AP Calculus AB and BC: Chapter 9 -Infinite Sequences and Series : 9.1 -Sequences and Series Study Notes

9.1 Sequences and Series

A sequence is an ordered list of numbers
$
a_1, a_2, a_3, \ldots, a_n, \ldots
$
whereas a series is an infinite sum of a list of numbers
$
a_1+a_2+a_3+\cdots+a_n+\cdots
$
Mathematically, a sequence is defined as a function whose domain is the set of positive integers.
If a sequence $\left\{a_n\right\}$ has the limit $L$, where $L$ is a real number, it is written as
$
\lim _{n \rightarrow \infty} a_n=L
$
and we say the sequence converges to $L$. If the sequence does not have a limit we say the sequence diverges.
Given a sequence of numbers of $\left\{a_n\right\}$, an expression of the form
$
\sum_{n=1}^{\infty} a_n=a_1+a_2+a_3+\cdots+a_n+\cdots
$
is an infinite series.
For the infinite series $\sum a_n$, the $\boldsymbol{n}$ th partial sum is given by
$
s_n=\sum_{i=1}^n a_i=a_1+a_2+a_3+\cdots+a_n
$

If the sequence of partial sums converges to a limit $L$, we say that the series converges, and its sum is $L$. If the series does not converge, we say that the series diverges.
The geometric series $a+a r+a r^2+\cdots+a r^{n-1}+\cdots$ converges to the sum
$
\sum_{n=1}^{\infty} a r^{n-1}=\frac{a}{1-\boldsymbol{r}}, \text { if }|r|<1 .
$
If $|r| \geq 1$, the series diverges.
A series is a telescoping series if it is in the form
$
\left(a_1-a_2\right)+\left(a_2-a_3\right)+\left(a_3-a_4\right)+\left(a_4-a_5\right)+\cdots \cdot
$
Since the $n$th partial sum of the series is $S_n=a_1-a_{n+1}$, a telescoping series converges if and only if $\lim _{n \rightarrow \infty} a_n=L$, where $L$ is a real number.
The sum of the series is $\boldsymbol{S}=\boldsymbol{a}_1-\boldsymbol{L}$.