%5En%2Fln(n)-converge-or-diverge.jpeg "(-1)^n/ln(n) Converge Or Diverge")
Determining the Convergence of the Series (-1)^n/ln(n)
The series (-1)^n/ln(n) is an alternating series. To determine its convergence, we can apply the Alternating Series Test.
Alternating Series Test
The Alternating Series Test states:
- Condition 1: The terms of the series must decrease monotonically. This means the terms get smaller as n increases.
- Condition 2: The limit of the terms must approach zero as n approaches infinity.
Let's examine our series in light of these conditions.
Applying the Alternating Series Test
Condition 1: The terms of the series are given by 1/ln(n). To see if they decrease monotonically, consider the derivative of this function:
d(1/ln(n))/dn = -1/(n*ln(n)^2)
Since the derivative is negative for n > 1, the function 1/ln(n) is decreasing monotonically. Therefore, Condition 1 is satisfied.
Condition 2: We need to find the limit as n approaches infinity:
lim (n -> infinity) 1/ln(n) = 0
This limit is indeed zero. Therefore, Condition 2 is also satisfied.
Conclusion
Since both conditions of the Alternating Series Test are satisfied, we can conclude that the series (-1)^n/ln(n) converges.
Important Note: While the series converges, it does not converge absolutely. To see this, consider the series formed by taking the absolute value of each term, i.e., 1/ln(n). This series diverges, as it is a p-series with p = 1, which is less than or equal to 1.