Applying the Integral Test
Use the Integral Test to dete...

Question

Applying the Integral Test

Use the Integral Test to determine if the series in Exercises 1–12 converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.

∑ (from n = 1 to ∞) 1 / n⁰·²

Accepted Answer

Hello. In this video we are going to be using the integral test to determine whether the series converges or diverges, and if we have a convergent series, we want to compute its sum. Now, the series given to us is the infinite series, starting from 2 and going to infinity of 1 divided by n raised to the power of 0.5. Now let's go ahead and review some conditions we need to meet in order to apply the integral test. Now the first thing we want to do is we want to go ahead and define a function in terms of X. We are simply going to take the argument of our summation and write that as a function of x so we can express the summation as 1 divided by x raised to the power of 0.5 and that can also be rewritten as x raised to the power of 1/2. Now, in order to apply the integral test, there are 3 conditions that we need to check. The first condition is to check whether the function is positive. Now here, the function will only be positive for all values of x that are greater than 0. But based on our summation, we are only working with values of x which are greater than or equal to 2. So since this function is always greater than 0, the function is going to be positive. The next thing we need to do Is checked for continuity. Now, again, because this function exists for all values of X that are equal to 0, this function is going to be continuous as well. And the last thing we need to check is we need to check and see if the function is decreasing. Now we can check this condition by taking the derivative of our function. The derivative of the function is going to be 1/22 multiplied by x raised to the power of -3 halves, but here this is going to be. This is going to be negative. For all values of x that are greater than or equal to 2, so the function is going to be decreasing. So now that we've satisfied these three conditions, we can take our function of X and form the For the form the improper integral. So the improper integral is going to be the integral from 2 to infinity. Of x raise the power of 1/2 d x. We will create a limit substitution for the infinite boundary, and we will state that the limit as b approaches infinity for the integral to raise to the power of b multiplied by x raise to the power of -1/2 d x. Now, if we were to derive or take the antiderivative of X to the negative12 power, we can rewrite this limit to be the limit as B approaches infinity of 2 multiplied by the square root of X, and this is going to be evaluated from 2 to B. Expanding this value will give us the limit as b approaches infinity of 2 multiplied by the square root of b minus 2 multiplied by the square root of 2. Now, once we apply our limit, we will be left with 2 multiplied by the square root of infinity minus 2 multiplied by the square root of 2, but this is just going to give us an output of positive infinity. Since we have an output of positive infinity, what does this tell us about the summation? Well, based off of the integral test, this summation is going to diverge because we have an infinite value. So the solution to this problem is that we have a divergent series from the integral test. And with that being said, the overall solution to this problem is going to be option D. So I hope this video helps you in understanding on how to approach this problem, and we will go ahead and see you all in the next video.

Applying the Integral TestUse the Integral Test to determine if the series in Exercises 1–12 converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.∑ (from n = 1 to ∞) 1 / n⁰·²

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Applying the Integral Test
Use the Integral Test to determine if the series in Exercises 1–12 converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.
∑ (from n = 1 to ∞) 1 / n⁰·²