 This lesson is on telescoping series. There are two types of series where you could actually find the sums. They are geometric series and telescoping series. Other series that we will be working on in the future will just be tested for convergence and divergence. But telescoping series and geometric series are the two that you can actually find those sums. What is a telescoping series? Well, when you have a series and you expand it out and the series collapses in the middle, which means terms are canceled out in the middle, you have a telescoping series. So what does a telescoping series look like? Well, in sigma notation, you could have something like n equals 2 to infinity of 1 over 2 times n plus 1 times n minus 1. In this case, you have a fraction. And if you look at that and try to expand it out, it will not do well, but you need to do something with it. And what we do is what we did with partial fractions, which means break it down into its a's and b's. So let's try one of these. Determine the sum of n equals 1 to infinity of 1 over n times n plus 1. So first thing we will do is break this down into its partial fractions. So we have 1 over n times n plus 1 is equal to a over n plus b over n plus 1. Multiplying out, we get 1 is equal to a times n plus 1 plus b times n. Putting in a 0 for n, we get 1 equals a. Putting in a negative 1 for n, we get 1 equals negative b or b equals negative 1. Putting this back into our series notation, we get a, which is 1 over n minus b, which is negative 1 over n plus 1. And of course, we're going to sum that from n is equal to 1 to infinity. Let's expand this. We get 1 minus 1 half when we put in a 1. When we put in the 2, we will get 1 half minus 1 third. When we put in the 3, we'll get plus 1 third minus 1 fourth, and this will go on forever and ever until we get to the end when we will get 1 over n minus 1 over n plus 1. As we go towards infinity, we can think of taking the limit on this. So we get the limit as n approaches infinity of all of this and what happens? Well, because this is a telescoping series, it collapses in the middle. So the 1 halves go out, the 1 thirds go out, the 1 fourths go out, all the way down the line. When we get to infinity, we have n approaching infinity, which makes this term zero and then the next term zero. So we actually end up with the sum of 1. And remember, this is the sum of this series. Let's look at another example. Determine the sum of n equals 1 to infinity of 4 over 4n minus 3 times 4n plus 1. Again, a telescoping series and we recognize it because we can put it into those partial fractions. So let's do the same with this one as we did before. We have 4 over 4n minus 3 times 4n plus 1 equals a over 4n minus 3 plus b over 4n plus 1. Simplifying this a little bit, we have 4 is equal to a times 4n plus 1 plus b times 4n minus 3. Substituting in negative 1 fourth for n, we get 4 equals a times 4 times negative 1 fourth plus 1 plus b times 4 times negative 1 fourth minus 3. And of course, the a term cancels out. So we get negative 4b is equal to 4 or b is equal to negative 1. Substituting in 3 fourths for n, we get 4. In this one, the b will cancel out and we'll get 4 equals a times 4 times 3 fourths plus 1. So that will cancel out and we'll get 4a is equal to 4 or a is equal to 1. Putting it back into our partial fractions, we get the sum of n equals 1 to infinity of 1 over 4n minus 3 minus 1 over 4n plus 1. Expanding out, we get 1 over, when we put the n equals 1 in, we get 4 times 1 is 4 minus 3 is 1 minus. Putting 1 in there, we get 1 fifth. That's our first set of terms plus substituting a 2 in. We get 1 fifth minus 1 ninth. Continue on forever. We'll get a collapse on this one too. And our last two terms will be 1 over 4n minus 3 minus 1 over 4n plus 1. Taking the limit as n approaches infinity on this, everything collapses. The ends become zeros again and we get the royal sum of 1. Hopefully these will go easily for you. Telescoping series is a really important series, just as important as geometric series when we are talking about series in calculus. This concludes your lesson on telescoping series.