$An$ explin function $is$ one which $is$ a sum $of$ multiples $of$ exponential functions. Such functions are

$of$ the form $f(x)=c_1{b}_1^x+c_2{b}_2^x+$cdots$+c_k{b}_k^x,$ where the $c_i$ are nonzero constants, and the $b_i$ are distinct

positive numbers with each $b_{i}1.$ For example, $f(x)=2(3^x)+7({0\mathrm{.}1}^x)-5({}^x)isan$ explin function.

$(a)$ For each $of$ the following series, use the $limit$ comparison test $to$ determine whether they

converge $or$ diverge:

$(i){\displaystyle \underset{n=1}{\overset{}{}}}\frac{0\mathrm{.}5(3^n)}{7^n+2(5^n)}$

$(ii){\displaystyle \underset{n=1}{\overset{}{}}}\frac{9^n+8^n-7^n}{8^n-7^n+6^n}$

$(iii){\displaystyle \underset{n=1}{\overset{}{}}}\frac{{0\mathrm{.}2}^n-{0\mathrm{.}1}^n}{7({0\mathrm{.}4}^n)+3({0\mathrm{.}1}^n)}$

$(b)Iff(x)$ and $g(x)$ are two explin functions, come $up$ with $(and$ prove$)$ a simple way $to$ determine

exactly when ${\displaystyle \underset{n=1}{\overset{}{}}}\frac{f(n)}{g(n)}$ converges $or$ diverges.