WEBVTT
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let's use the root test to determine whether or not
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the Siri's convergence. So let's call this our Anne
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then the root tests requires that we look at the
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limit as n goes to infinity and to root absolute
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value. And so, in our case, let
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me go ahead and use this fact from algebra that
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you could always write. The end through is just
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the rational exponents one over it. So that's what
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I'LL do here with this and through. So let's
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write the limit and goes to infinity So already I'll
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just write that is the one over any year Oops
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! Sorry about that. Using this fact over here
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and then absolute value of And so go ahead and
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take the absolute value of this fraction. The numerator
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just becomes a one the denominator natural log of end
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to the end power. And here I don't need
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absolute value anymore in the denominator, because natural log
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of end, it's bigger than zero. If Anne
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is bigger than or equal to two. And that's
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exactly the case that we're in for this problem.
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So now I could we could think of the inside
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here. This could be really in as well.
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One is equal to one to the end power.
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And we could write one over once in the end
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over natural life and to the end as one over
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natural log of end all to the end power.
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Then we're still raising that to the one over.
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And and now we use the fax from algebra that
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if you haven't exponents eight of the bee and if
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you raise that to another exponents E, then that's
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the same is just raising a to the B times
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C. So here this will be my baby and
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my cease all just multiplied those together. And when
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we do that, they just cancel auto one and
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we're left over with one over natural log. Since
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natural law goes to infinity, the fraction goes to
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zero. That's less than one. So we conclude
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that the Siri's convergence by the root test