Answer
converges
Work Step by Step
Given $$\sum_{n=2}^{\infty}(1-\sqrt{1-\frac{1}{n^{2}}}) $$
Since
\begin{aligned}
a_{n}&=1-\sqrt{1-\frac{1}{n^{2}}}\\
& =1-\sqrt{\frac{n^{2}-1}{n^{2}}}\\
& =1-\frac{\sqrt{n^{2}-1}}{n}\\
& =\frac{n-\sqrt{n^{2}-1}}{n}\\
& =\frac{n-\sqrt{n^{2}-1}}{n} \times \frac{n+\sqrt{n^{2}-1}}{n+\sqrt{n^{2}-1}}\\
&= \frac{1}{n^{2}+n \sqrt{n^{2}-1}}
\end{aligned}
Compare with the convergent series $\sum \frac{1}{n^2}$:
\begin{align*}
\lim_{n\to \infty } \frac{a_n}{b_n} &= \lim_{n\to \infty } \frac{n^2}{n^{2}+n \sqrt{n^{2}-1}}\\
&=1
\end{align*}
Then $\sum_{n=2}^{\infty}(1-\sqrt{1-\frac{1}{n^{2}}})$ converges.
