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Numerade Educator



Problem 32 Medium Difficulty

Test the series for convergence or divergence.

$ \displaystyle \sum_{n = 1}^{\infty} \frac {(n!)^n}{n^{4n}} $




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Video Transcript

in this problem, we have to test a Siris for the convergence or divergence. But the only tricky thing is is we're not told what test use. We kind of have to use our knowledge about Siri's to determine the test. So let's first review the Siri's that were given were given this Siri's from n equals one to infinity of in factorial raise to the end all over and raised to foreign. So let's apply the test for divergence. And let's just see what we get the test for Divergent says that we can take the limit is then approaches Infinity of Ace event. Well, that would be the limit as an approaches infinity of in factorial over end to the fourth, all raised to the end. Now this is a little bit tricky. We have to be comfortable with factorial, so let's just simplify what we have. We'll get The limit is un approaches infinity of an times n minus one times n minus, two times and minus three all over and to the fourth times and minus four Factorial race to the end and then we could simplify that we get the limit is n approaches infinity of one minus one over N times one minus two over end times one minus three over in, and we'll multiply that by end minus four. Factorial all race to the end. Well, now we're in a position that we can plug in infinity to our limit. When we do that, we'll get one minus zero times one minus zero times, one minus zero times infinity all raised to infinity. Well, that's clearly infinity. So that means that we found that our limit approached infinity. So our Siri's is divergent by the test for divergence. So I hope that this problem helped you understand a little bit more about Siri's and how to choose a test and also how we can go about showing the diversions of a Siri's by using the test for divergence.