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Numerade Educator



Problem 67 Hard Difficulty

If the $ n $th partial sum of a series $ \sum_{n = 1}^{\infty} a_n $ is
$ s_n = \frac {n - 1}{n + 1} $
find $ a_n $ and $ \sum_{n = 1}^{\infty} a_n. $


$\sum_{n=1}^{\infty} a_{n}=1$


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Video Transcript

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