%5En%2Fn-converge-or-diverge.jpeg "(-1)^n/n Converge Or Diverge")
Does the Series (-1)^n/n Converge or Diverge?
The series (-1)^n/n is a classic example in calculus that demonstrates the concept of conditional convergence. Let's explore why.
Understanding the Series
The series (-1)^n/n represents an alternating series, meaning its terms alternate in sign. The series can be written as:
-1 + 1/2 - 1/3 + 1/4 - 1/5 + ...
The Alternating Series Test
We can use the Alternating Series Test to determine if the series converges. This test has two conditions:
- The terms must decrease in absolute value: |(-1)^n/n| = 1/n, and 1/n is a decreasing sequence as n increases.
- The limit of the terms must approach zero: lim (n→∞) 1/n = 0.
Since both conditions are met, the Alternating Series Test tells us that the series (-1)^n/n converges.
Conditional Convergence
However, if we consider the absolute value of the terms, we get the series 1/n. This series is the harmonic series, which is known to diverge.
Therefore, the series (-1)^n/n converges conditionally, meaning it converges only because of the alternating signs. It would diverge if we were to consider the absolute values of the terms.
Conclusion
The series (-1)^n/n is a fascinating example of conditional convergence. While it converges due to the alternating nature of its terms, the series of its absolute values diverges. This highlights the importance of understanding different types of convergence in calculus.