I have the following problem
$A=\sum_{n=1}^{\infty}\frac{2n+7^n}{2n+6^n}$
... and I'm trying to figure out if it is convergent or divergent using the Comparison Test.
A similar problem:
$D = \sum_{n=1}^{\infty}\frac{4n+5^n}{4n+8^n}$
... is solved by using the Comparison Test like so:
$\frac{4n+5^n}{4n+8^n}<\frac{5^n+5^n}{4n+8^n}<\frac{2\cdot 5^n}{8^n}$
and since:
$2\sum _{n=1}^{\infty }\frac{5^n}{8^n}$
... is a convergent Geometric Series, $D$ converges too.
If I try to do something similar to the original problem, $A$, I get the following:
$B=\sum _{n=1}^{\infty }\frac{7^n}{6^n}$
... is a divergent geometric series.
Therefore, I believe I need to use the comparison test to show that:
$A>B$
... this time, and I will be able to conclude that $A$ is divergent as well. If I try doing that, though, I run into a problem:
$\frac{2n+7^n}{2n+6^n}>\frac{7^n}{2n+6^n}<\frac{7^n}{6^n}$
Am I comparing it to the wrong function? Can I not use the Comparison Test on this problem and instead need to use the Integral Comparison Test?