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Numerade Educator



Problem 42 Medium Difficulty

Determine whether the series is convergent or divergent. If it is convergent, find its sum.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {e^n}{n^2} $




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Video Transcript

let's go ahead and determine whether or not the serious conversions and if it's convergent, will go ahead and find the summer as well. So for this problem here, the first thing we might look at to see is if it's geometric and in this one we can just we could see that the answer is no. You cannot write this problem in the form. Eight times are the end You were A and R don't depend on him. So let's try another test here. Let's try the test for diverges, which is in the which is stated in the section in the book. So here we look at the women and goes to infinity of either the end over and square. Now I claim that this is will be equal to infinity and a way to see this hour. This technically the limit. Right now, the ways rent is of the form infinity over infinity and from complice. You might remember this and your previous calculus course. This is what the author calls the indeterminate form and usually use low Patel's rule, which is perfectly fine here. But some people would not use Low Patel when you're using the value of that because these functions here or not continuous because you're not using all roll numbers, you're only using natural numbers. So what some people would do is they would love replace and with X That way you're dealing with the rial valued functions here and these are continuous these air differential so you can go ahead and use love with house rules. So let me just write. I'Ll hop there for low Patel. We take the limit X goes to infinity derivative on the top to rivet upon the bottle. This is still of the forum infinity over infinity, So use low Patel again. So take the derivative of top and bottom once more and then we get either the ex over to. And then now this is either the X divided by a number and this is equal to infinity. So therefore we have that the limit exist. So let's just write it this way. Since the limit of N goes to infinity of either then over and square is not equal to zero. We read it this way equals infinity. So it exists but is not equal to zero. The Siri's and equals want to infinity E n over and square diverges by the diversions test. This is a very important test, and that's my final answer.