The Conditional Convergence of the Series -1^n 1/n

Understanding the behavior of series in mathematics is crucial for analyzing the convergence properties of various mathematical series. This article delves into the analysis of the series given by the terms -1^n cdot frac{1}{n}. We will utilize the Alternating Series Test (also known as the Leibniz test) to determine the conditions under which this series converges and whether this convergence is absolute or conditional.

Analysis with the Alternating Series Test

To analyze the convergence of the series given by the terms -1^n cdot frac{1}{n}, we employ the Alternating Series Test. An alternating series has the form:

[sum_{n1}^{infty} (-1)^n a_n]

where (a_n) is a sequence of positive terms. According to the Alternating Series Test, the series converges if the following two conditions are satisfied:

Step 1: Check the Conditions for the Alternating Series Test

Decreasing Terms

The sequence (b_n frac{1}{n}) is positive and decreasing for (n geq 1). As (n) increases, (b_n) decreases because the reciprocal of a larger number is smaller.

Limit Condition

We need to check the limit condition:

[lim_{n to infty} b_n lim_{n to infty} frac{1}{n} 0]

As (n) approaches infinity, the value of (frac{1}{n}) approaches 0, satisfying the limit condition.

Step 2: Conclusion from the Alternating Series Test

Since both conditions of the Alternating Series Test are satisfied, the series (sum_{n1}^{infty} -1^n cdot frac{1}{n}) converges.

Step 3: Determine the Type of Convergence

To determine the type of convergence, we need to check if the series converges absolutely. For absolute convergence, we consider the series:

[sum_{n1}^{infty} left(-1^n cdot frac{1}{n}right) sum_{n1}^{infty} frac{1}{n}]

This is the harmonic series:

[sum_{n1}^{infty} frac{1}{n}]

The harmonic series is known to diverge. Therefore, the series does not converge absolutely.

Final Conclusion:

Since the series converges but does not converge absolutely, we conclude that:

[sum_{n1}^{infty} -1^n cdot frac{1}{n} text{ is conditionally convergent.}]

Value of the Series

The series converges to (ln(2)). This is a known result derived from the Taylor series expansion of (ln(1 x)), specifically (ln(1 1) ln(2)).

Therefore, the series (sum_{n1}^{infty} -1^n cdot frac{1}{n}) is conditionally convergent to (ln(2)).