8.(8pts) Determine whether the series is absolutely convergent, conditionally convergent, or divergent. Circle the answer.
(1) $\sum_{n=1}^{\infty} \ln \left(1+\frac{1}{n}\right)$
(abs. converges, cond. converges, diverges.)
(2) $\sum_{n=1}^{\infty}(-1)^{n} \frac{n+1}{n \sqrt{n}}$
(abs. converges, cond. converges, diverges.)
(3) $\sum_{n=1}^{\infty}(-1)^{n} \sin \left(\frac{1}{n^{2}}\right)$
(abs. converges, cond. converges, diverges.)
(4) $\sum_{n=1}^{\infty}(-1)^{n} \frac{2021^{n}}{2022^{n}+2020^{n}}$
(abs. converges, cond. converges, diverges.)

Step 1 ：First, we will determine the convergence of each series using the appropriate tests:

Step 2 ：For series (1), we can use the comparison test.

Step 3 ：For series (2), we can use the limit comparison test with the series \(\sum_{n=1}^\infty \frac{1}{\sqrt{n}}\).

Step 4 ：For series (3), we can use the alternating series test.

Step 5 ：For series (4), we can use the ratio test.

Step 6 ：Applying these tests, we find the following results:

Step 7 ：The series \(\sum_{n=1}^\infty \ln \left(1+\frac{1}{n}\right)\) diverges.

Step 8 ：The series \(\sum_{n=1}^\infty(-1)^{n} \frac{n+1}{n \sqrt{n}}\) is conditionally convergent.

Step 9 ：The series \(\sum_{n=1}^\infty(-1)^{n} \sin \left(\frac{1}{n^{2}}\right)\) is absolutely convergent.

Step 10 ：The series \(\sum_{n=1}^\infty(-1)^{n} \frac{2021^{n}}{2022^{n}+2020^{n}}\) is absolutely convergent.

Step 11 ：\(\boxed{\text{Final Answer:}}\)

Step 12 ：\(\boxed{\text{(1) diverges}}\)

Step 13 ：\(\boxed{\text{(2) conditionally convergent}}\)

Step 14 ：\(\boxed{\text{(3) absolutely convergent}}\)

Step 15 ：\(\boxed{\text{(4) absolutely convergent}}\)