17–22. Integral Test Use the Integral Test to determine whethe...

Question

17–22. Integral Test Use the Integral Test to determine whether the following series converge after showing that the conditions of the Integral Test are satisfied.

∑ (k = 1 to ∞) 1 / (∛(5k + 3))

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. Use the integral test to determine whether the series, the sum evaluated from N equals 2 to positive infinity of 1 divided by the square root of 4 N + 7 converges or diverges. 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 use the integral test to determine whether or not this particular series converges or diverges. So our final answer will be either A converges or B diverges. So with that in mind, let us first recall what the integral test is and apply it to this particular prompt. So as we should recall, using the integral test, we will let F of X be positive. So I'll put a little plus sign in quotes. So we'll let F of X be positive, continuous, and decreasing. So I'll put an arrow pointing down for decreasing, and it'll be decreasing for X is greater than or equal to N. If the integral evaluated from N to positive infinity of F of X D X converges, meaning it's finite, then the series, which will be the sum evaluated from lowercase n is equal to N to positive infinity of F of N, specifically lowercase n. So F of lowercase n converges. If the integral diverges, however, the series will diverge also. So with that in mind, our first step that we need to take in order to solve this problem is we need to define the corresponding function. So let us state that F of X is equal to 1 divided by the square root of 4 X + 7, which will be equal to parentheses 4 x + 7 to the power of -12. Where F of X in this case is continuous, positive, and it is, let's make a note, and it is decreasing for X is greater than or equal to 2. Which means as a result, the integral test will apply to this particular problem. Our second step now is we need to set up the improper integral, so we need to take the integral evaluated from 2 to positive infinity of parentheses 4 X plus 7 to the power of -15 DX. Fantastic. So moving right along here, our third step now is we need to recall and use the substitution method. So let us state that you will be equal to 4 X plus 7, which will mean that DU is equal to 4DX, which will mean that DX is equal to DU divided by 4. And let us note that when X is equal to 2, that will mean that U is equal to 4 multiplied by 2, which will be added to 7, so you. Let's take a moment here to pause. So for the condition when X is equal to 2, that will mean that U is equal to 4 multiplied by 2, plus 7, which will be equal to 15. Fantastic. And when X approaches positive infinity, so once again, as X approaches positive infinity, that will mean that you will also approach positive infinity. So, including this information and keeping it in mind, the integral will become the integral evaluated from 2 to positive infinity of parentheses 4 X + 7 to the power of negative 1/2. DX, which will be equal to 1/4 multiplied by the integral evaluated from 15 to positive infinity of U to the power of negative 12 DU. Fantastic. So, our 4th step now is we need to integrate. So when we integrate, we will find that the integral of you to the power of, so we're gonna take the integral. Of U to the power of negative 12 DU will be equal to 2 multiplied by the power of 12 plus C. And when we multiply by 14th, we will find that our answer will be 1/4 multiplied by parenthesis of 2 multiplied by U to the power of 1/2 plus C, which will be equal to 1/2 multiplied by u to the power of 1/2 plus it's going to be plus C divided by 4, which will be equal to 1/2 multiplied by U to the power of 1/2 plus C. Where C will be equal to C divided by 4. Fantastic. Our 5th step now is we need to evaluate the limit, so we need to take the integral evaluate it from 2 to positive infinity of parentheses 4 X plus 7 to the power of -12 DX, which will be equal to the limit as B approaches positive infinity of 1/2 multiplied by the square root of U, which will be evaluated from 15 to. 4B + 7. Which will be equal to the limit as B approaches positive infinity of. One half multiplied by the square root of 4B plus 7 minus 12 multiplied by the square root of 15. Which will mean that the limit as B approaches positive infinity of 12 multiplied by the square root of 4B plus 7 will be equal to positive infinity. Fantastic. So, because it's equal to positive infinity, the integral diverges, so let's make a note of that, it diverges. And because of this fact, We can stay in our 6th and final step. That when we make our final conclusions, when we use the integral test, we need to recall and note that based on everything we know about the integral test, in conclusion, since the improper integral diverges for this particular case, the series also diverges. So that will mean that the sum evaluated from N equals 2 to positive infinity of 1 divided by the square root of. 4 N + 7 will also diverge, so it diverges, and that is our final answer. Hooray, we did it. So going back to look at our multiple choice answers, our final answer once again, without a doubt, will have to be multiple choice answer letter B, which is diverges. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

17–22. Integral Test Use the Integral Test to determine whether the following series converge after showing that the conditions of the Integral Test are satisfied.
∑ (k = 1 to ∞) 1 / (∛(5k + 3))

Key Concepts

Integral Test

Conditions for Applying the Integral Test

Evaluating Improper Integrals

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