Test the following series for convergence. \(\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{2}}\)

An alternating series is a series where the signs of the terms alternate between positive and negative. To test for convergence of an alternating series, we use the Alternating Series Test, also known as Leibniz's Test. This test helps us determine whether an alternating series converges by checking a couple of important criteria.
For an alternating series of the form \(\sum_{n=1}^{\text{inf}} (-1)^n a_n\), the conditions we need to verify are:

• The absolute value of the terms \( a_n \) should be decreasing: \( a_{n+1} < a_n \).
• The terms must approach zero as \( n \rightarrow \text{inf} \): \( \rightarrow 0 \).
  

If both of these conditions are met, the series converges.
In our example series \(\sum_{n=1}^{\text{inf}} \frac{(-1)^{n}}{n^2} \), the terms are \( \frac{1}{n^2} \), which are positive and decreasing, and they approach zero as \( n \rightarrow \text{inf} \). Thus, this series converges by the Alternating Series Test.

Leibniz's Test is another name for the Alternating Series Test, named after the mathematician Gottfried Wilhelm Leibniz. This test is crucial in determining the convergence of alternating series.
Leibniz's Test requires us to:

• Confirm that the terms \( a_n \) form a decreasing sequence.
• Verify that the limit of \( a_n \) as \( n \rightarrow \text{inf} \) is zero.
  

Let's revisit our series example \(\sum_{n=1}^{\text{inf}} \frac{(-1)^{n}}{n^2} \):
Here, \( a_n \) is given by \( \frac{1}{n^2} \).
1. The terms \( \frac{1}{n^2} \) are positive and decreasing because as \( n \) gets larger, \( n^2 \) increases, making \( \frac{1}{n^2} \) smaller. \( \frac{1}{(n+1)^2} < \frac{1}{n^2} \).2. The limit of \( \frac{1}{n^2} \) as \( n \rightarrow \text{inf} \) is zero, satisfying the criteria of the test.
Therefore, using Leibniz's Test, the alternating series \(\sum_{n=1}^{\text{inf}} \frac{(-1)^n}{n^2}\) converges.

Convergence criteria are the rules that determine whether a series converges or not. For alternating series, the primary tool for checking convergence is the Alternating Series Test.
In this test, the convergence criteria are:

• The sequence of terms \( a_n \) must be positive and monotonically decreasing.
• The limit of the terms \( a_n \) as \( n \rightarrow \text{inf} \) must be zero.
  

Applying these criteria to the series \(\sum_{n=1}^{\text{inf}} \frac{(-1)^n}{n^2} \), we check that:

• \( a_n \) = \( \frac{1}{n^2} \), which is positive for all \( n \).
• The sequence \( a_n \) = \( \frac{1}{n^2} \) is decreasing because as \( n \) increases, \( \frac{1}{n^2} \) gets smaller.
• We have \( \text{lim}_{n \to \text{inf}} \frac{1}{n^2} = 0 \).

Thus, since these criteria are met, the series converges.
Understanding these criteria is essential for analyzing the convergence of alternating series.