Does $\sum \frac{(n+4^n)}{n+6^n}$ converge or diverge?

Question

The Question

Does $\sum \frac{(n+4^n)}{n+6^n}$ converge or diverge?

Please note I have no knowledge of Alternating Series, Ratio and Root tests, Power Series, or Taylor and McLaurin Series.

My Work

The integral test didn't really appeal to me cause I wasn't sure how to take that integral. If that is the easiest approach and someone wants to help me with the integral then please show me the way.

By taking the simple Divergence test (the limit as n goes to infinity) I got a result of $0$ which unfortunately tells me nothing.

I can't really find a good value for my Comparison tests either. Can anyone help with any of these methods?

Answer 1

Hint: $\frac{n+4^n}{n+6^n} \leq\frac{n+4^n}{6^n} \leq 2(\frac{2}{3})^n$

The last is because $n < 4^n$. Can you do the rest?

Answer 2

HINT: d'Alembert test (do you know it?) and $\frac23<1$.

Without d'Alembert test: $$ \frac{(n+4^n)}{n+6^n}\leq\frac{2\cdot4^n}{6^n}, $$ for large $n$ (in fact: always), and geometric sequence you should know.

Answer 3

Them limit will be similar to the limit of them largest growing terms. That would be $(4/6)^n$.