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 (√(n² + n) − n)]

Accepted Answer

Determine whether the series sum equals 1 to infinity of -1 to the n multiplied by the square root of n2 + 2n minus n converges or diverges. We have two possible answers, of course, being converging or diverging. Now, to solve this, we'll actually make use of the divergence test, which tells us if the limit as M approaches infinity of a subm is not equal to 0 or it does not exist, this series diverges. So let's identify our parts of our series. You know, AN is just -1 ton multiplied by the square root of n2 + 2N minus n. That means BN is just the square root term, square root. Of n2 + 2 n minus n. Now, let's make this a little easier to work with by rationalizing. We can rationalize by multiplying by its conjugate square root of n2 + 2 n + n. In a fraction. Now we can simplify this. This just simplifies to give us. 2 n divided by the square root n2 + 2 n + n. Now from here we can do some more simplification. Let's divide the numerator and denominator by n. So when we do this, we have 2 N divided by N, all divided by the square root of N2 divided by N2, plus 2N. Divided by n 2. Plus n divided by n. Now we put N2d inside the square root because it was a square rooted term. This just simplifies to give us 2 divided by the square root. Of 1 + 2 divided by n Plus one Now we have a form we can work with. And we can take the limit. As n approaches infinity of 2 divided by the square root of 1 + 2 divided by n + 1. This gives us 2 divided by the square root of 1. Plus 0. Plus one Which is simplifies to give us 2 divided by 2 or 1. We then have the limit. As n approaches infinity of b_n equals 1. By our divergence test, this limit does not approach 0. So we can note that down. As in process infinity, bn is not equal to 0. And so, because we have an alternating term as well, these values alternate between 1 and -1, meaning that this limit actually does not exist. We can then say the limit as it approaches affinity of a sub n. Doesn't exist, which means this diverges based on our divergence tests. This means then that the answer to our problem is answer B. 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 (√(n² + n) − n)]

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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 (√(n² + n) − n)]