WEBVTT
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Let's first note that the terms that we're dealing with
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their positive. So first, let's look at the
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one plus co signed in. We know that CO
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sign of end is less than or equal to one
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favor than or equal to negative one. So if
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we add one so all signs of this inequality here
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we have zero less than or equal to one plus
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coz I less than or equal to two. So
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that shows that our numerator is positive and we know
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he's always positive. The reason I'm checking this is
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because if you like to use the comparison test,
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you have to make sure that your Siri's has on
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ly positive terms. And that's what we have here
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. A n bigger than zero or equal to That's
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fine, just no negatives. So now let's go
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ahead and use comparison his test here. So I
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know one plus co sign in is less than or
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equal to two. So this tells me that our
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Siri's is less than or equal to to overeat of
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the end. All I'm doing here is just using
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this inequality that one plus coastline is less than or
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equal to two and then we can rewrite this.
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Pull out the two and then we could write.
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This is one over e to the end. This
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is a geometric Siri's. We see that our equals
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one over e rough estimates of this would just be
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a third. But all that matters is that it's
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less than one an absolute value, one over three
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, more or less, and that's less than one
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. So any time it's geometric series. Satisfied this
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? We know that it converges. Therefore, since
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we have a Siri's with positive terms and it's founded
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above by a convergence here ese by the comparison test
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our series, which is one plus co sign and
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over eat of the end. Also convergence okay,
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and that's your final answer.