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 ∞) 1 / (√k × e^(√k))

Accepted Answer

Below there. Today we're going to 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 1 divided by the power of 3 divided by 2? Multiplied by e to the power of M divided by 2 converge. Awesome. So it appears for this particular prompt we're asked to determine whether or not this series that is provided to us by the prom itself converges. So will it be either A, yes, the series converges, or B, no, the series diverges. Awesome. So now that we know what we're ultimately trying to solve for, our first step that we need to take in order to solve this problem is we need to recognize the type of series that we are dealing with. So right off the bat, so just by looking at the series, we need to be able to recall and recognize that each term is positive. And because that each, so each term is positive, this positive value, we can recall and use the comparison tests. So we can use the direct comparison or the limit comparison. So with that in mind, our second step is we need to compare with a simpler series. So we need to notice that E to the power of M divided by 2 grows much faster than any polynomial M to the power of 3 divided by 2. So that will mean we can safely say that 0 is less than 1 divided by M to the power of 3 divided by 2 multiplied by E to the power of M divided by 2. Fantastic. And this is less than 1 divided by E to the power of M divided by 2. Fantastic. So with that in mind, We now need to note that for our 3rd step, we need to check the simpler series. So we need to take the sum that is evaluated at or rather evaluated from M equals 1 to positive. of 1 divided by e to the power of M divided by 2, which will be equal to the sum evaluated from M equals 1 to positive infinity of parentheses 1 divided by the square root of E to the power of M. And we need to be able to recall and recognize that this is a geometric series with a ratio R that is equal to 1 divided by the square root of E, which is going to be less than 1. And this geometric series with the absolute value of R will be less than 1, will converge, which I'll call C for short for converge. So, with that in mind, our 4th and final step is we need to recall and apply the direct comparison test. And since 0 is less than 1 divided by M to the power of 3 divided by 2, so 3 divided by 2 multiplied by E to the power of M divided by 2. Is less than 1 divided by E to the power of M divided by 2. And that the sum of 1 divided by E to the power of M divided by 2. Converges. It therefore follows that the sum. Evaluated from M is equal to 1 to positive infinity of 1 divided by M to the power of 3 divided by 2 multiplied by. E to the power of M divided by 2. Converges due to the direct comparison, so M divided by 2. So that means that this series converges, which I'll call C once again for short, by the direct comparison, which I'll call DC. So it can once again, long story short, This series converges by the direct comparison, so that means based on the information that we've collected together, our final answer will be multiple choice answer letter A, which once again is yes, the series converges, and that's it. That is 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 k = 1 to ∞) 1 / (√k × e^(√k))

Key Concepts

Convergence of Infinite Series

Comparison and Limit Comparison Tests

Exponential and Root Functions in Series Terms

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