Problem
Question 9 (4 points) \( \quad \) srved From the series listed below, select ALL convergent series. \[ \sum_{n=1}^{\infty} \frac{(-1)^{n}}{\sqrt{n+3}} \] \[ \sum_{n=1}^{\infty} \frac{(2 n+1) !}{3^{n}} \] \[ \sum_{n=1}^{\infty} \frac{\sqrt{n^{2}+1}}{n^{3}+5} \] \[ \sum_{n=1}^{\infty}(-1)^{n} \frac{n}{n+1} \]
Solution
Step 1 ：Apply the alternating series test to \( \sum_{n=1}^\infty \frac{(-1)^n}{\sqrt{n+3}} \).
Step 2 ：Determine the convergence of \( \sum_{n=1}^\infty \frac{(2n+1)!}{3^n} \) using the ratio test.
Step 3 ：Examine the convergence of \( \sum_{n=1}^\infty \frac{\sqrt{n^2+1}}{n^3+5} \) by comparing with a known convergent series.
Step 4 ：Apply the alternating series test to \( \sum_{n=1}^\infty (-1)^n \frac{n}{n+1} \).
