Question 9 (4 points) srved
From the series listed below, select ALL convergent series.
$\sum_{n = 1}^{\infty} \frac{(- 1)^{n}}{\sqrt{n + 3}}$
$\sum_{n = 1}^{\infty} \frac{(2 n + 1)!}{3^{n}}$
$\sum_{n = 1}^{\infty} \frac{\sqrt{n^{2} + 1}}{n^{3} + 5}$
$\sum_{n = 1}^{\infty} (- 1)^{n} \frac{n}{n + 1}$

Answer

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Answer

Apply the alternating series test to $\sum_{n = 1}^{\infty} (- 1)^{n} \frac{n}{n + 1}$ .

Steps

Step 1 ：Apply the alternating series test to $\sum_{n = 1}^{\infty} \frac{(- 1)^{n}}{\sqrt{n + 3}}$ .

Step 2 ：Determine the convergence of $\sum_{n = 1}^{\infty} \frac{(2 n + 1)!}{3^{n}}$ 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 $\sum_{n = 1}^{\infty} (- 1)^{n} \frac{n}{n + 1}$ .