The series you're referring to seems to be a bit unclear. However, if you're referring to the series ∑ ((-1)^n * n^2) / n^n (from n=1 to ∞), we can test for convergence or divergence using the Alternating Series Test.

Step 1: Identify bn. In an alternating series of the form ∑ (-1)^n * bn, bn is the part of the series that does not include the (-1)^n term. So in this case, bn = n^2 / n^n.

Step 2: Check if bn+1 ≤ bn for all n. This means that the terms of the series are decreasing. In this case, (n+1)^2 / (n+1)^(n+1) ≤ n^2 / n^n. This is true for all n ≥ 1.

Step 3: Check if the limit as n approaches infinity of bn is 0. In this case, the limit as n approaches infinity of n^2 / n^n is 0.

Since both conditions of the Alternating Series Test are met, the series converges.

Question

The series you're referring to seems to be a bit unclear. However, if you're referring to the series ∑ ((-1)^n * n^2) / n^n (from n=1 to ∞), we can test for convergence or divergence using the Alternating Series Test.

Step 1: Identify bn. In an alternating series of the form ∑ (-1)^n * bn, bn is the part of the series that does not include the (-1)^n term. So in this case, bn = n^2 / n^n.

Step 2: Check if bn+1 ≤ bn for all n. This means that the terms of the series are decreasing. In this case, (n+1)^2 / (n+1)^(n+1) ≤ n^2 / n^n. This is true for all n ≥ 1.

Step 3: Check if the limit as n approaches infinity of bn is 0. In this case, the limit as n approaches infinity of n^2 / n^n is 0.

Since both conditions of the Alternating Series Test are met, the series converges.

Knowee AI · Accepted Answer

The series you're referring to seems to be a bit unclear. However, if you're referring to the series ∑ ((-1)^n * n^2) / n^n (from n=1 to ∞), we can test for convergence or divergence using the Alternating Series Test.

Step 1: Identify bn. In an alternating series of the form ∑ (-1)^n * bn, bn is the part of the series that does not include the (-1)^n term. So in this case, bn = n^2 / n^n.

Step 2: Check if bn+1 ≤ bn for all n. This means that the terms of the series are decreasing. In this case, (n+1)^2 / (n+1)^(n+1) ≤ n^2 / n^n. This is true for all n ≥ 1.

Step 3: Check if the limit as n approaches infinity of bn is 0. In this case, the limit as n approaches infinity of n^2 / n^n is 0.

Since both conditions of the Alternating Series Test are met, the series converges.

Test the series for convergence or divergence.∞(−1)nn2nn = 1Identify bn.

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