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√10)]

Accepted Answer

Determine whether the series, the sum, as in equals 1 to infinity of -1 to the n. Multiplied by the nth root of 7. Converges or diverges. And we have two possible answers being converging or diverging. Now to solve this, we'll make use of the divergence test. Which tells us if the limit as M approaches infinity of the series AM is not equal to zero, or it doesn't exist, the series diverges. Now for this, we can identify what our general term is. We'll actually use n since we have n in our original. So a_n equals -1 to the n multiplied by the nth root of 7. So we need to determine this limit and see if it's equal to 0 or something else. So we need to calculate the limit of this roots. To do this, we'll take BN, which is just our AN without the negative1 to the end, or the oscillating term. That is the nth root of 7. We then take the limit as n approaches infinity of the n through of 7. We can then rewrite this. We have 7 raised to the 1. Divided by n. Now if we were to use our limit and plug in infinity, we get 7 rays to the 1 divided by infinity, or 7 rays to the 0, which is just 1. So, based on Are a subn We have -1 raises the n multiplied by the limit. Which is just one. Now here, -1, raised to the N is actually going to oscillate between -1 and 1. For the even numbers, it's positive, and for the odd numbers, it's negative. Because this is always oscillating, it never approaches a specific value. Which means that this limit. As in approaches infinity of a sub n. Does not exist. We can then say that this limit as in the approach infinity of a subn is then not equals to zero. By our tests, this diverges. This means 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⁺¹ (ⁿ√10)]

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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√10)]