WEBVTT
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were given that the end partial some of this Siri's
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here is s end On one hand, we know
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that s end by definition is the sum of the
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A's starting in this case that one So a one
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all the way up to and for this particular sum
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were given that this is equal to and minus one
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over and plus one. So there's two things to
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find here. First, we'LL find an so a
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n we can get from doing sn and then minus
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s n minus one. So the way to see
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this if you do s and minus s and minus
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one then this is just a one all the way
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up to a n. So this is for us
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and and then ascend minus one is when you only
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add from one all the way up to and minus
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one. So when we subtract, we're on ly
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able to cancel the first and minus one numbers there
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, and we're loved over with Anne. So that
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verifies this equation over here and now. Since we
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know we have a formula for both of these terms
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, on the right hand side, we could just
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use this formula here that's given to us for us
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. N oh, so plug in as then and
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minus one over and plus one. And then over
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here you were place s and with us and minus
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one. So wait, so that gives us and
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minus two over end. And we could either stop
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right there or go ahead and just combine distractions,
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Sosa, and minus two and plus one. And
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this can be simplified a little bit. So what
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we get here? We have I'm squared minus end
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. And then we have this minus. And then
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here, this is and square minus one and minus
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two. So said Frank of that. So pushed
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the negative sign through the parentheses and you have negative
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and square minus negative end and then minus negative too
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allover and and plus one still cancel is much as
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we can, and we're left over with two over
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end times and plus one. And this is the
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formula for our A n. So that was the
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left hand's left most side over here. We started
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with an A and we ended with two over end
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times and plus one. So that's the first part
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of this question. Now, the answer The second
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part. We have to go ahead and find this
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whole infinite sum here. So let's go to the
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next page for this We had s and equals and
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minus one over and plus one. Now we want
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to find the sum, want to infinity and so
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we can write. This is Lim. Let's say
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Que goes to infinity of S k. So remember
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this is just the This is by definition let me
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use the same notation so I could rewrite The infinite
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sum is a limit And then I could use the
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definition of S k to replace this kid Partial some
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here with S K. However, we have a
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formula for S K over here. So this is
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just a limit Es que goes to infinity of K
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minus one over Kaye plus one. And this limit
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is just equal to one. So this answers the
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question What is the infinite? Some of the Siri's
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And it's just one. So that's our final answer