WEBVTT
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Let's show that this Siri's converges for all ex.
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Some party, eh? With use the ratio test
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for this is our r A n So the ratio
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test for requires us to look at a N plus
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one over a end absolute value of that and then
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take the limit is and goes to infinity. So
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here will have so that a m plus one of
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the top and then they end in the bottom.
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Now we should go ahead and flip that red fraction
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over and multiply dividing by a fraction here. So
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we'LL have X and plus one over X n and
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then in factorial over and plus one that's for real
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. So here's the key observation here. You just
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look at this fraction. So using the definition of
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the factorial function, she write that out, You
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could see that we could cancel the first and batters
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and you just slept over with one over and plus
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one. So here will have just one ex leftover
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eggs on top and plus one in the bottom.
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But here, for any real Number X that we
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have excessive fixed number, it's not a limit here
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. The end is in the limit, and this
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will always go to zero due to that denominator.
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So you could think of this is being X over
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infinity. But X is not infinity or minus Infinity
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X is a real number, so we have zero
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, which is less than one. So we conclude
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that it converges for all ex, and that was
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the majority of the work here because part B will
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follow from the test for divergence. So this is
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a corollary of the test for diversions. So sorry
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here, mixing on my letters. The test for
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diversion says if the Siri's converges, then the limit
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of the and has to go to zero. So
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that's just the theorem in the book. But in
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our case, the limit of a end based on
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our definition of an this is just a limit as
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n goes to infinity X to the end over and
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factorial. So you go back and erase that.
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That's and factorial there kind of running out of room
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here. Scylla meeting There's my in factorial and then
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by the serum above that were citing, This has
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to be zero and that our deduction in part B
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and that completes our answer