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JH
Numerade Educator

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Problem 20 Easy Difficulty

Use the Ratio Test to determine whether the series is convergent or divergent.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {(2n)!}{(n!)^2} $

Answer

divergent

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Video Transcript

accused the ratio test to determine whether the Siri's convergence. So let's look at this term here and Jeff two in Factorial, Divided by and Factorial Square. Now the ratios has suggests that we look at the limit and goes to infinity absolute value and plus one over. And let me go back there. Live a little sloppy a n plus one over an. Now we can drop absolute value here because AM is positive. So the numerator let me do this in red. This is too, and plus two so noticed that if you increase enter and plus one, this is two, one plus two and then we have factorial and then n plus one factorial and that's clear divided by hand. Now let's go ahead and simplify this a little bit. So this term up here, this to one plus two factorial I can write that is two in factorial two in factorial times two in plus one times two, one plus two. That's this term right here. And then I could also bring this and Fats for real square up into the numerator. And then I'm left over with here I penis two in factorial and then I have n plus one Factorial saw me, right? That is and factorial times and plus one and then that square. So this is corresponding to this term. Over here correspond Silvestre. And of course, here I should write that limit on the front somewhere living is and goes to infinity. Let me bring this down to the bottom left. So here we should cancel out as much as we can. We see that we can take off those to inventory ALS here we have in fact, Foreal Square that'LL cancel off with this and factorial that's being squared. And then if we walk around, we see that in the numerator we still have two in plus one two in plus two In the denominator, you have n plus one square. This limit is equal to four. Just look at the leading terms up these quadratic CE. This is bigger than one. So we conclude that the Siri's from one to infinity two in factorial over. In fact, for the square damages one by the ratio test. And that's our unless, sir