[Math] Is the series $\sum 5^n/(4^n+3)$ convergent or not

Question

Is the series $\sum \dfrac{5^n}{4^n +3}$ convergent or divergent?

Actually I have started the problem by applying the root test but got stuck so as what to do with the denominator.

If there is any other method to apply please tell me that.

Answer 1

As a sequence, $$\frac{5^n}{4^n+3}\to\infty \mbox{ as }n\to\infty. $$ In particular, $$\lim_{n\to\infty }\frac{5^n}{4^n+3}\neq 0.$$ Therefore, the series $$\sum_{n=1}^\infty \frac{5^n}{4^n+3}$$ diverges by divergence test.