Answer
$a_{k}=\frac{5^k + k}{k! +3}$
$b_{k}=\frac{5^k + k}{k!}$
$=\lim\limits_{k \to infinity} \frac{a_{k}}{b_{k}}$
$=\lim\limits_{k \to infinity}\frac{5^k + k}{k! +3} \times \frac{k!}{5^k + k}$
$=\lim\limits_{k \to infinity} \frac{k!}{k!+3}$
Take k! as a common factor from both numenator & denominator
$=\lim\limits_{k \to infinity} \frac{1}{1+3/k!}$
put the limit....
$=\frac{1}{1+0}$
$=1$
So we can conclude that given series converges by limit comparison test....
Work Step by Step
First take the given series...
Then take another series from given series by eliminating constants....
then apply the limit comparison test...
After applying, we obtain a finite answer...
by the conditions of limit comparison test when we get a finite answer,the series must be a convergent series....
