00:01
Let's determine whether the series converges or diverges by expressing sn as a telescoping sum.
00:10
And then if it converges, we'll go ahead and find that sum as well.
00:15
So sn, recall by definition, is just the sum of the first n numbers, where your an is this thing that's being added up.
00:30
So let's just go ahead and write this is a limit.
00:34
K goes to infinity.
00:36
And then we have the sum from n equals 1 to k e 1 over n minus e 1 over n plus 1 now this term here in the parentheses this whole sum this is s k that's the sum from 1 all the way up tn just as it's defined so let's telescope let's do a telescoping sum only inside of the parentheses and then after that we'll come out here and take the limit and that should answer our question so let's write this so now let's go ahead and start adding so plug in n equals 1 first that's our starting point so e to the 1 over 1 minus e to the 1 over 2 this is the first term after plugging in n equals 1 and then next you increase n by 1 so n is now 2 n equals 2 there maybe one more term in this direction so e so now n is 3 so plug that in and you might see some cancellation happening there already so we would keep going in this direction so put some dots there to indicate that and then eventually at the very end when we plug in n equals k we'd have e to the 1 over k minus e to the 1 over k plus 1 but in order to maybe help you see the pattern here it might be best to write a term before n equals k.
02:28
So the term right before the last one, k minus 1.
02:38
And by writing that, you can see some more cancellation over here.
02:42
The e to the 1 over k will cancel with negative e to the 1 over k.
02:46
So now let's see how much we could cancel.
02:50
E to the 1 over 1.
02:52
This doesn't cancel with anything because all the remaining denominators are larger than 1...