Questions: Determine whether the alternating series sum from n=2 to infinity of (-1)^(n+1) * 3/(5(ln n)^2) converges or diverges. Choose the correct answer below and, if necessary, fill in the answer box to complete your choice. A. The series converges by the Alternating Series Test. B. The series does not satisfy the conditions of the Alternating Series Test but converges because it is a geometric series with r= . C. The series does not satisfy the conditions of the Alternating Series Test but diverges because it is a p-series with p= . D. The series does not satisfy the conditions of the Alternating Series Test but diverges by the Root Test because the limit used does not exist. E. The series does not satisfy the conditions of the Alternating Series Test but converges because it is a p-series with p= .

Determine whether the alternating series sum from n=2 to infinity of (-1)^(n+1) * 3/(5(ln n)^2) converges or diverges.

Choose the correct answer below and, if necessary, fill in the answer box to complete your choice.
A. The series converges by the Alternating Series Test.
B. The series does not satisfy the conditions of the Alternating Series Test but converges because it is a geometric series with r= .
C. The series does not satisfy the conditions of the Alternating Series Test but diverges because it is a p-series with p= .
D. The series does not satisfy the conditions of the Alternating Series Test but diverges by the Root Test because the limit used does not exist.
E. The series does not satisfy the conditions of the Alternating Series Test but converges because it is a p-series with p= .

Transcript text: Determine whether the alternating series $\sum_{n=2}^{\infty}(-1)^{n+1} \frac{3}{5(\ln n)^{2}}$ converges or diverges. Choose the correct answer below and, if necessary, fill in the answer box to complete your choice. A. The series converges by the Alternating Series Test. B. The series does not satisfy the conditions of the Alternating Series Test but converges because it is a geometric series with $r=$ $\square$ . C. The series does not satisfy the conditions of the Alternating Series Test but $\square$ diverges because it is a $p$-series with $p=$ . D. The series does not satisfy the conditions of the Alternating Series Test but diverges by the Root Test because the limit used does not exist. E. The series does not satisfy the conditions of the Alternating Series Test but converges because it is a $p$-series with $p=$ $\square$ .

Solution

Solution Steps

To determine whether the given alternating series converges or diverges, we can use the Alternating Series Test (Leibniz's Test). The Alternating Series Test states that an alternating series $\sum (-1)^{n} b_n$ converges if the following two conditions are met:

The sequence $b_n$ is monotonically decreasing.
$\lim_{n \to \infty} b_n = 0$.

For the given series $\sum_{n=2}^{\infty}(-1)^{n+1} \frac{3}{5(\ln n)^{2}}$, we need to check these conditions for $b_n = \frac{3}{5(\ln n)^{2}}$.

Solution Approach

Verify that $b_n = \frac{3}{5(\ln n)^{2}}$ is monotonically decreasing.
Check that $\lim_{n \to \infty} b_n = 0$.

Step 1: Check Monotonicity

We need to verify if the sequence $ b_n = \frac{3}{5(\ln n)^{2}} $ is monotonically decreasing. We compute the derivative:

\[ b_n' = \frac{d}{dn} \left( \frac{3}{5(\ln n)^{2}} \right) = -\frac{6}{5n(\ln n)^{3}} \]

Since $ n > 1 $ and $ \ln n > 0 $, it follows that $ b_n' < 0 $. Therefore, $ b_n $ is monotonically decreasing.

Step 2: Check the Limit

Next, we check the limit of $ b_n $ as $ n $ approaches infinity:

\[ \lim_{n \to \infty} b_n = \lim_{n \to \infty} \frac{3}{5(\ln n)^{2}} = 0 \]

Since both conditions of the Alternating Series Test are satisfied (i.e., $ b_n $ is monotonically decreasing and $ \lim_{n \to \infty} b_n = 0 $), we conclude that the series converges.