Absolute and Conditional Convergence
Which of the series...

Question

Absolute and Conditional Convergence

Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.

∑ (from n = 1 to ∞) [(-1)ⁿ⁻¹ / (n² + 2n + 1)]

Accepted Answer

Determine whether the series, the sum as it equals 4 to infinity, of -1 to the n divided by n2 minus 4N + 4, converges or diverges. And we have two possible answers being converges or divergence. Now, to solve this, we'll make use of the alternating series test. This tells us, suppose we have a series. AN of the form -1 nbn or -1 n + 1bn, where BN is greater than equals 0 for all n. Then if the limit as it approaches infinity of bn equals 0 and B7 is a decreasing sequence. A subn is convergent. In other words, we need to check if these two conditions hold. So, let's identify our term. So, we can say B7. As equals 21 divided by n 2 minus 4 n + 4, because this follows the form of an alternating series. We can actually simplify this denominator by noticing it can be factored. We can write this as 1 divided by n minus 2 squad. Now we can apply our test. To do this, let's first determine if this is decreasing. Now, as N increases, the denominator N minus 22 increases, which means the fraction overall decreases. So this is overall decreasing. Now we just need to check if the limit goes to 0. We have the limit as n approaches infinity of 1 divided by n minus 2 square. We'd get 1 divided by infinity, which just equals 0. Because The polynomial function goes to infinity. We confirm this condition. And then we can state that by the alternating series test, this Convergence Which means the answer to our problem is answer A. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

Absolute and Conditional ConvergenceWhich of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.∑ (from n = 1 to ∞) [(-1)ⁿ⁻¹ / (n² + 2n + 1)]

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Absolute and Conditional Convergence
Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.
∑ (from n = 1 to ∞) [(-1)ⁿ⁻¹ / (n² + 2n + 1)]