Answer
See below.
Work Step by Step
Examples of a divergent series:
$\displaystyle \sum_{n=1}^{\infty}1=1+1+1+1+\cdots$
$\displaystyle \sum_{n=1}^{\infty}(-1)^{n}=-1+1-1+1-1+1-1+\cdots$
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n},\qquad $ (the harmonic series)
This includes any series for which the terms do not approach 0 as $ n\rightarrow\infty$.
Examples of convergent series:
See example 5 in the section on series (the telescoping series),
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n(n+1)}=1$
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{2^{n}}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}\quad=1$
Later, we will see the power series, Taylor and MacLaurin series.
There is a multitude of each type of series, as terms of series may be combined, separated, extracted and recombined.
