Problem 44 Medium Difficulty

Determine whether the series is convergent or divergent by expressing $ s_n $ as a telescoping sum (as in Examples 8). If it is convergent, find its sum.
$ \displaystyle \sum_{n = 1}^{\infty} \ln \frac {n}{n + 1} $

Answer

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Video Transcript

let's determine whether the Siri's is conversion or diversion by using the telescoping song. So hear what we'LL do is rewrite this summation as a limit. So instead of looking at the whole sum all at once, let's on Lee Look at that's two s k. So remember, But this is by definition of the Siri's. It's just a limit of the partial sums and you let the index go to infinity. Now, in our case, we have Lim kay goes to infinity s k means the sum from our starting point, which is one all the way up to Kay and then we have Ellen and then end over and plus one in this case is actually going to be better if we want to telescope usual odd properties to rewrite. This is natural log of end minus Ellen of plus one. So this these are the terms that were adding up. So before we take the limit, let's ignore that for a moment and on Lee, focus on what's inside the apprentices. And then after we simplify, we'LL come back over here and take the limit. So me come down here, Lim Now let's go ahead and right this some oak. So you start with n equals one. We have Ellen one minus ln of one plus one. Ellen to minus ln three Ellen three minus Ellen for and so on. We see the pattern and then we keep going. And for example, when we plug it K minus one. And then finally you were plugging and equals K. And let's go ahead and try to simplify this. Well, I want of one. We know that zero and then we have natural log up to, but that cancels with positive. Eleanor too. Negative, Ellen on three Positive, Eleanor, three and so on. All the negatives will cancel out with a positive even all the way up into you. Getto negative, natural lot of kay. Here's positive, natural on cocaine And even this term right here. This would cancel with the term right before it, because the term right before it would be Ellen of K minus two, minus Ellen of K minus one. And then that would cancel with this. So everything cancels except this very last term here with the negative sign. So we have let lim Kay goes to infinity of negatives. Natural log of K plus one. But we know from the O. R picture of the natural log that as thie input is larger, the graph goes to infinity. We have a negative sign here, so this is going to a negative infinity. So the limit is not a real number. And since the limits not a real number, this Siri's is diversion. Any time the limit is infinity minus infinity or even if the limit doesn't exist in any of those cases, we say that the syriza's divergent and that's our final answer by using the telescoping method from example eight.