(-1)^n/ln(n) Converge Or Diverge

3 min read Jun 16, 2024
(-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.

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