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 + √n)]

Accepted Answer

Determine whether the series, the sum as in equals 1 to infinity of -1 to the n, divided by 2 plus n 1/3 converges or diverges. And we just have two possible answers being converging or diverging. Now, to solve this, we'll make use of the alternating series tests. This tells us to suppose that we have a series, AN, where AN is 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 affinity of the series BN equals 0, and BN is a decreasing sequence. The sum of a subn is convergent. Now, to solve this, we first check if this follows the form, and it does because we have -1 raced to the end. So we can then denote B_N as 1 divided by 2 plus n raised to the 1/3 power. Now we need to verify our conditions. So let's check if this is a decreasing sequence first. We can do this by checking if. Bn + 1 is less than or equals to bn. So Let's find bn + 1. Wish this is 1 divided by 2 plus n + 1 raised to the 1/3, and check if it is less than or equals to 1 divided by 2 plus n to the 1/3. Now, based on what we have here, we know that n + 1 1/3 is going to be larger than n 1/3. But since we have the reciprocal, when we put this in the denominator, this makes this a smaller value. In other words, BN + 1 is less than BN, so this check is confirmed. Now we just need to check if the limit. As n approaches infinity of 1 divided by 2 plus n to the 1/3 is equal to 0. Now we know that in 1/3 is actually the cube root function, and this function grows towards infinity when n approaches infinity. So therefore, We have 1 divided by infinity, or 0. So this limit does indeed go to zero. That means both conditions are confirmed, and we can say that this converges by the alternating series tests. Meaning our answer 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 + √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 / (1 + √n)]