How do you determine the convergence or divergence of #Sigma ((-1)^nsqrtn)/root3n# from #[1,oo)#?

Answer 1

The series is not convergent as it does not satisfy Cauchy's necessary condition.

If we write the general terms of the series as:

#a_n = sqrt(n)/root(3)n = n^(1/2)/n^(1/3) = n^(1/2-1/3) = n^(1/6)#

we can see that:

#lim_(n->oo) a_n = oo#

so the series cannot converge.

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Answer 2

To determine the convergence or divergence of the series ( \sum_{n=1}^{\infty} \frac{(-1)^n \sqrt{n}}{\sqrt{3n}} ) from ( n = 1 ) to ( \infty ), we can use the Alternating Series Test.

First, note that the sequence ( a_n = \frac{\sqrt{n}}{\sqrt{3n}} = \frac{1}{\sqrt{3}} ) is a decreasing sequence that approaches zero as ( n ) approaches infinity. Additionally, the series alternates in sign with ( (-1)^n ).

By the Alternating Series Test, since the terms are decreasing in magnitude and approach zero as ( n ) approaches infinity, and the series alternates in sign, the series ( \sum_{n=1}^{\infty} \frac{(-1)^n \sqrt{n}}{\sqrt{3n}} ) converges.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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