a) For all $n > 1, 0 \le \frac{\sin^2(n)}{n^2} \le \frac{1}{n^2}$, and the series $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges, so by the Comparison Test, the series $\sum_{n=1}^{\infty} \frac{\sin^2(n)}{n^2}$ converges.
Correct
b) For all $n > 1, 0 \le \frac{\arctan(n)}{n^3} < \frac{\pi}{2n^3}$, and the series $\frac{\pi}{2}\sum_{n=1}^{\infty} \frac{1}{n^3}$ converges, so by the Comparison Test, the series $\sum_{n=1}^{\infty} \frac{\arctan(n)}{n^3}$ converges.
Correct
c) For all $n > 2, \frac{\log(n)}{n^2} > \frac{1}{n^2} \ge 0$, and the series $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges, so by the Comparison Test, the series $\sum_{n=2}^{\infty} \frac{\log(n)}{n^2}$ converges.
Incorrect
d) For all $n > 2, 0 \le \frac{n}{n^3 - 6} < \frac{2}{n^2}$, and the series $2\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges, so by the Comparison Test, the series $\sum_{n=1}^{\infty} \frac{n}{n^3 - 6}$ converges.
Correct
e) For all $n \ge 3, 0 \le \frac{1}{n^2 - 8} < \frac{1}{n^2}$, and the series $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges, so by the Comparison Test, the series $\sum_{n=1}^{\infty} \frac{1}{n^2 - 8}$ converges.
Incorrect
f) For all $n > 1, \frac{n}{2 - n^3} < \frac{1}{n^2}$, and the series $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges, so by the Comparison Test, the series $\sum_{n=1}^{\infty} \frac{n}{2 - n^3}$ converges.
Correct