11–86. Applying convergence tests Determine whether the follow...

Question

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 1 to ∞)(⁵√k) / ⁵√(k⁷ + 1)

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Does the series, the sum evaluated from M equals 1 to positive infinity of the cube root of M divided by the cube root of M the power of 5 + 4 converge. Awesome. So it appears for this particular problem we're asked to determine whether or not this particular series either A converges or B diverges. So now that we know we're ultimately trying to solve our noting that we're focusing on this specific series that's provided to us, our first step that we need to take in order to solve this particular problem. is we need to be able to recognize that when we look at the series that we have positive term series that we're dealing with and because we have a positive term series, that means we can recall and use the limit comparison test with a known P series is a convenient way to go to solve this problem. So as we should recall, the limit comparison test for positive sequences, for example, A subscript M and B subscript M, if L lowercase l is equal to the limit of, or rather the limit as M approaches positive infinity of a subscript M divided by B subscript M. Fantastic. So, that means that it exists and that 0 is less than L and L is less than positive infinity. So then that will mean that the sum of a subscript M and the sum of subscript B, so the sum of subscript. B subscript M, either both converge or both diverge. Awesome. So now we can perform our step by step solution to solve this problem, starting with our first major step that we need to take to solve this problem. we need to write the general term. So the general term, a subscript M is equal to the cube root of M divided by the cube root of M to the power of 5 + 4, which is equal to m of 1/3 divided by parenthesis M 5 + 4 of 1/3. Fantastic. So our second step now is we need to factorm the power of 5 out of the cube root in the denominator, so we need to take parentheses m of 5 + 4 of 1/3, which is equal to m of 5 divided by 3 multiplied by parentheses 1 + 4 divided by m of 5. To the power of 1/3. Fantastic. We are on a roll here. So, that will mean, and we can go ahead and write for our third step that A subscript M. is equal to m of 13 divided by m of 5/3 multiplied by parenthesis of 1 + 4 divided by the power of 5. Fantastic. All to the power of 1/3, which will be equal to M to the power of -4 divided by 3, multiplied by 1 + 4 divided by M to the power of 5, all to the power of -13. Fantastic. So that will mean that our fourth step is we need to compare B subscript M is equal to 1 divided by the power of 4 divided by 3. Awesome. So it's N to the power of 4 divided by 3. We have to be specific here. Fantastic. So now we need to compute the limit in order to do that, so we need to take the limit as M approaches positive infinity of a subscript M divided by B subscript M. Which is equal to the limit, as M approaches positive infinity of 1 + 4 divided by M to the power of 5. Fantastic. All to the power of -13, which will be equal to 1 + 0 to the power of negative 1/3, which will be equal to simply 1. And as we can see, due to our result of getting an answer of 1, the limit is finite and positive. So that will mean for our 5th and final step we need to make our final conclusions. So since the sum of B subscript M is equal to the sum of 1 divided by M to the power of 4 divided by 3. It is a convergent, which I'll just say C for short, it is a convergent P series, which I'll call PS for short, PS for short. So because the sum of B subscript M equals the sum of 1 divided by M to the power of 4 divided by 3, this is without a doubt a convergent P series, meaning that P is equal to 4 divided by 3, which is greater than 1. So that means that the limit comparison test applies that the sum of a subscript M also converges. So that means that our final answer without a doubt will have to be converges. Hooray, we did it. So converges is our final answer. So looking at our multiple choice answers, the correct answer without a doubt. Has to be multiple choice answer letter A, which once again is converges. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)(⁵√k) / ⁵√(k⁷ + 1)

Key Concepts

Convergence of Infinite Series

Comparison Test for Series

Asymptotic Behavior and Simplification of Terms

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