Question

Let \( a_k = (-1)^{k-1} \), \( s_n = a_1 + a_2 + \cdots + a_n \), and \( \sigma_n = \frac{1}{n} (s_1 + s_2 + \cdots + s_n) \), where \( k, n \in \mathbb{N} \). Then

\[ \lim_{n \to \infty} \sigma_n = \text{.................} \text{ (correct up to one decimal place)}. \]

Hint:

For alternating series, the partial sums tend to oscillate around a value. The average of these sums tends to the limit of the series.

Answer 1

Correct Answer: 0.4

Step 1: Understanding the terms.
We are given that \( a_k = (-1)^{k-1} \), so the sequence \( a_1, a_2, a_3, \dots \) alternates between 1 and -1. Thus, the partial sums \( s_n \) will alternate between two values as \( n \) increases. The formula for \( \sigma_n \) is an average of these partial sums.

Step 2: Behavior of \( s_n \) and \( \sigma_n \).
The sequence \( s_n \) oscillates, and as \( n \to \infty \), the average \( \sigma_n \) tends to a limit. By the alternating series behavior, we find that the average value of \( s_n \) approaches 0 as \( n \) increases. Thus, the limit of \( \sigma_n \) as \( n \to \infty \) is 0.

Step 3: Conclusion.
\[ \lim_{n \to \infty} \sigma_n = \boxed{0}. \]