Answer
a) A convergent sequence is one that reaches a finite limit.
Examples:
1) $a_{n}= \frac{n+2}{2n-1}$
2) $a_{n}= \frac{n}{n+1}$
b) A divergent sequence is one that does not reach a finite limit.
Examples:
1)$a_{n}=n^{2}$
2)$a_{n}=\frac{n^{2}}{n+1}$
Work Step by Step
a) $a_{n}= \frac{n+2}{2n-1}$
$a_{1}=3$
$a_{3}=1$
$a_{1000}=0.50125$
$a_{10000}=0.500125$
$a_{n}= \frac{n}{n+1}$
$a_{1}=0.5$
$a_{3}=0.6666$
$a_{1000}=0.9990$
$a_{10000}=0.99990000999$
b)
$a_{n}=n^{2}$
$a_{1}=1$
$a_{3}=9$
$a_{1000}=1000000$
$a_{9999}=99980001$
$a_{n}=\frac{n^{2}}{n+1}$
$a_{1}=1$
$a_{3}=0.9$
$a_{1000}=999.000999001$
$a_{9999}=9998.0001$
