Answer
Converges
Work Step by Step
Th. 9.13 Limit Comparison Test:
If $a_{n} > 0, b_{n} > 0$, and $\displaystyle \lim_{n\rightarrow\infty}\frac{a_{n}}{b_{n}}=L,$
where $L$ is finite and positive, then
$\displaystyle \sum_{n=1}^{\infty}a_{n}$ and $\displaystyle \sum_{n=1}^{\infty}b_{n}$ either both converge or both diverge.
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Let $a_{n}=\displaystyle \frac{5}{4^{n}+1}$.
...If the numerator had a factor $4^{n}$
the limit would be easily found, which is why we observe
$\displaystyle \sum_{n=1}^{\infty}b_{n}=\sum_{n=1}^{\infty}\frac{1}{4^{n}} =\displaystyle \sum_{n=1}^{\infty}(\frac{1}{4})^{n}$
which is a convergent geometric series.
Test the hypothesis of the theorem:
$\displaystyle \lim_{n\rightarrow\infty}\frac{a_{n}}{b_{n}}$ =$\displaystyle \lim_{n\rightarrow\infty}\frac{\frac{5}{4^{n}+1}}{\frac{1}{4^{n}}}=\lim_{n\rightarrow\infty}\frac{5\cdot 4^{n}}{4^{n}+1}$
... divide both numerator and denominator with $ 4^{n}$...
$=\displaystyle \lim_{n\rightarrow\infty}\frac{5}{1+\frac{1}{4^{n}}}=5=L$ (finite and positive)
So,
$\displaystyle \sum_{n=1}^{\infty}a_{n}$ converges as well (since they both converge).
