Answer
Diverges
Work Step by Step
Given $$\sum_{n=5}^{\infty} \frac{1}{\sqrt{n-4}}$$
Since $f(n)= \frac{1}{\sqrt{n-4}}$, is positive, decreasing and continuous for $n\geq5$, then by using the integral test
\begin{aligned} \int_{5}^{\infty} f(n) dn &=\lim _{t \rightarrow \infty} \int_{5}^{t} \frac{1}{\sqrt{n-4}}d n \\
&=\left.\lim _{t \rightarrow \infty}\frac{2}{3} (n-4)^{3/2}\right|_{5} ^{t} \\
&=\frac{2}{3}\lim _{t \rightarrow \infty} (t-4)^{3/2}-1 \\ &=\infty\end{aligned}
Thus the series diverges.
