For the questions that ask about positive numbers: zero $is$ not considered a positive number, $tobe$ clear. $We$ have seen some families $of$ sequences and series that can $be$ thought $ofas$ being $pa-$

rameterized $by$ a parameter $k.$ For example, the family $of$ geometric series $is{\displaystyle \underset{n=1}{\overset{}{}}}a{k}^{n-1}.$

Setting $k$ equal $to$ any real number gives a geometric series. For each series $in$ the family,

$kis$ fixed and changing $k$ gives a different series $in$ the same family.

Consider the family $of$ sequences given $by\{(1+\frac{1}{n}{)}^k\}.$ This sequence converges $to$

$1$ for every positive value $k.Ifwe$ create a new sequence $by$ replacing the parameter

$k$ with the index variable $n,we$ get the sequence $\{(1+\frac{1}{n}{)}^n\}$ which still converges, but

now $toe,$ not $1.$

The following questions ask for families $of$ sequence$\frac{s}{s}$eries where the convergence prop$-$

erties change when the parameter $is$ replaced $by$ the index variable.

$(a)(6$ points$)$ Write down three sequences $or$ series parameterized $byk$ that converge

for every positive value $ofk,$ but $do$ not converge when the parameter $kis$ replaced

$by$ the index variable $n.$

For each, justify convergence and divergence and for the convergent sequenc$\frac{e}{s}$eries,

find the $limitin$ terms $ofkif$ possible. $(b)(6$ points$)$ Write down three sequences $or$ series parameterized $byk$ that $do$ not

converge for any positive value $ofk,$ but $do$ converge when the parameter $kisre-$

placed $by$ the index variable $n.$

For each, justify convergence and divergence and find the $limitof$ the convergent

sequenc$\frac{e}{s}$eries $if$ possible.