Answer
Converges, by Th. 9.12
Work Step by Step
Th. 9.12 Direct Comparison Test:
Let $0 < a_{n} \leq b_{n}$ for all $n$.
1. If $\displaystyle \sum_{n=1}^{\infty}b_{n}$ converges, then $\displaystyle \sum_{n=1}^{\infty}a_{n}$ converges.
2. If $\displaystyle \sum_{n=1}^{\infty}a_{n}$ diverges, then $\displaystyle \sum_{n=1}^{\infty}b_{n}$ diverges.
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Observe that
$\displaystyle \frac{4^{n}}{5^{n}+3} < (\frac{4}{5})^{n} $ for $ n \geq 1$
$ a_{n}=\displaystyle \frac{4^{n}}{5^{n}+3},\qquad b_{n}=(\frac{4}{5})^{n}\cdot$
$ 0 < a_{n} \leq b_{n}$
(1). Since $\displaystyle \sum_{n=1}^{\infty}(\frac{4}{5})^{n}$ converges
(geometric series$, |r|=\displaystyle \frac{4}{5} < 1$),
then $\displaystyle \sum_{n=1}^{\infty}a_{n} =\displaystyle \sum_{n=1}^{\infty}\frac{4^{n}}{5^{n}+3}$ also converges