%5En%2Fn-converge.jpeg "(-1)^n/n Converge")
Does the Series (-1)^n/n Converge?
The series (-1)^n/n is a classic example of an alternating series, meaning its terms alternate in sign. To determine if it converges, we can use the Alternating Series Test:
Alternating Series Test
An alternating series of the form ∑ (-1)^n*a_n converges if:
- a_n > 0 for all n
- a_n ≥ a_(n+1) for all n (terms are decreasing)
- lim (n→∞) a_n = 0 (limit of terms is zero)
Applying the Test to (-1)^n/n:
- a_n = 1/n is positive for all n.
- 1/n ≥ 1/(n+1) for all n. The terms are decreasing.
- lim (n→∞) 1/n = 0.
Since all conditions are met, the Alternating Series Test tells us that the series (-1)^n/n converges.
What does convergence mean?
Convergence means that the series has a finite sum. While we can't easily calculate the exact sum of this series, we know it exists because the terms become increasingly smaller and eventually approach zero.
Understanding the Behavior:
The series (-1)^n/n oscillates back and forth as n increases. However, the oscillations become smaller and smaller because the denominator is increasing. This leads to the series converging to a specific value.
Important Note:
Even though the series converges, it does not converge absolutely. This means that the series ∑ |(-1)^n/n| = ∑ 1/n diverges. This is known as conditional convergence.
Conclusion
The series (-1)^n/n converges due to the Alternating Series Test. It converges conditionally, meaning it converges only because of the alternating signs. The series does not converge absolutely.