Question

To test this series for convergence

Answer 1

r should be 4/7. You're comparing the given series to a geometric series that converges. By the limit comparison test, you have

$\displaystyle\lim_{n\to\infty}\frac{\frac{4^n+7}{7^n}}{\frac{4^n}{7^n}}=\lim_{n\to\infty}\frac{4^n+7}{4^n}=\lim_{n\to\infty}\left(1+\frac7{4^n}\right)=1$

and since this limit is positive and finite, and the series you're comparing to is convergent, then the first series must also be convergent.