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 j = 2 to ∞)1 / (j ln¹⁰j)

Accepted Answer

Below 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 N equals 4 to positive infinity of 1 divided by N multiplied by the natural law to the power of 5 of N converge or diverge? Awesome. So it appears for this particular prompt we're asked to take the series that is provided to us by the prom itself and we're asked to determine whether or not this particular series converges or diverges. So it's either going to be A converges or B diverges as our final answer. So, with that in mind, Let us recall and apply the interval test to this particular problem to help us easily solve. For this series, for whether or not it's going to converge or diverge, and an easy way to determine whether or not it will converge or diverge is to use the integral test. And as you should recall, Using the interval test, we will let F of X be positive, so I put a positive symbol and quotation marks. So let F of X be positive, continuous, and decreasing, so I have an arrow pointing down. Decreasing for X is greater than or equal to N. If the integral evaluated from N to positive infinity of F of XDX converges, or rather it is finite, Then the series, the sum as so we're gonna have the sum evaluated from lowercase n is equal to N to positive infinity of F of lowercase n converges. However, if the integral diverges, the series will also diverge. So with that in mind, Our first step that we need to take is we need to define the corresponding function. So let us state that F of X is equal to 1 divided by X, multiplied by the natural log to the power of 5 of X, where X is greater than or equal to 4. And we need to note that F of X for this particular problem for this case is positive, continuous, and is decreasing on X is greater than or equal to 4. So X is greater than or equal to 4, and it's decreasing on this interval. So, That means based on these conditions, the integral test does indeed apply. So with that in mind, our second step is we need to write the improper interval. So we need to note that that as the interval test tells us to consider, the integral evaluated from 4 to positive infinity of F of X D X is equal to the integral evaluated from 4 to positive infinity of DX divided by X multiplied by the natural log to the power of 5 of X. If this integral converges, the series will also converge. Or converges. However, if it diverges, the series as a result will also diverges or will diverge. So once again, quick and easy to remember, if this particular integral converges, then that will mean our series converges. However, if this integral diverges, that will mean as a result, the series will diverge. So now that we have that in our mind, we need to do our next step, which is step 3, which is we're gonna have to recall and use substitution. So let us state that U is equal to the natural log of X, which will mean that D U is equal to DX divided by X, so Once again, we're using the substitution method, so U is gonna be equal to the natural log of X, and DU is gonna be equal to DX divided by X, which will mean that DX is equal to X multiplied by DU. Then that will mean that DX divided by X will be equal to DU, and that will mean that DX divided by X multiplied by parenthesis of the natural log of X to the power of 5 will be equal to DU divided by U to the power of 5. Fantastic. So now we need to change the limits. So when X is equal to 4, that will mean that U is equal to the natural log of 4. And when X approaches positive infinity, that will mean that you will also approach positive infinity. So with this knowledge in mind, our integral will become the integral evaluated from the natural log of 4 to positive infinity. Of DU divided by U to the power of 5. Fantastic. Our fourth step is we now need to integrate, so we need to take the interval of U to the power of -5 D U is equal to the power of -4 divided by -4 plus C, which will be equal to -1 divided by 4 multiplied by U to the power of 4 plus C, fantastic. So that will mean that our 5th step Is that we need to evaluate our improper integral, so we need to take the integral evaluated from the natural log of 4 to positive infinity of U to the power of -5. DU will be equal to the limit as B approaches positive infinity of square brackets of -1 divided by 4 multiplied by U to the power of 4. Evaluated from the natural log of 4 to B. Which will be equal to the limit as B approaches positive infinity of parentheses, negative 1 divided by 4 multiplied by B to the power of 4 + 1 divided by 4 multiplied by the natural log to the power of 4 of 4. Fantastic. So that will mean that the limit as B approaches positive infinity of -1 divided by 4 multiplied by B, the power 4 will be equal to 0. So that will mean that as a result, the integral equals 1 divided by 4 multiplied by parenthesis of the natural log of 4 to the power 4 is less than positive infinity. So, our 6th and final step is we need to make our final conclusions. Once again, keeping in mind the integral test. So our final conclusions by the integral test reveals that since the integral converges, that means the series also converges. So that will mean that the sum. Evaluated from N equals 4 to positive infinity of 1 divided by N multiplied by parenthesis of the natural log of N to the power of 5. Also converges. And that's it. That is our final answer. Hooray, we did it. So going back to look at our multiple choice answers, our final answer without a doubt will have to be multiple choice answer letter A, which once again is converges. And that's it. That's our final answer. 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 j = 2 to ∞)1 / (j ln¹⁰j)

Key Concepts

Convergence of Infinite Series

Integral Test for Convergence

Behavior of Logarithmic Functions in Series

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