 this video we'll talk about geometric series. Now we're going to talk about sums and it's a geometric series because series is when you add up all the terms. Well there's a nice little derivation for this one too but I'm just going to teach it to you giving you the formula. Basically it ends up to be a sub, take our first term and then we have one minus r to the n over one minus r. So let's see what we know. Partial sum. I have a sub one is equal to two. I also need my r so r is going to be equal to eight divided by two equal four. Thirty two divided by eight equal four. So r is equal to four and so we found r and we're ready to go. The only other thing we need to know is that n is equal to seven so s sub seven is equal to a sub one which is two times one minus my ratio which is four to the n happens to be seven and then I have one minus my common ratio which is four. Well if I do that I'm just going to come to my calculator and simplify the top but I know this is going to be negative three and if I do the top I have two times the parentheses one minus four caret seven close my parentheses but it's not 22 and enter and we have negative three two seven six six divided by negative three so I get a positive number and then just let my calculator do the divided by negative three and we get 10,922. I sum up all those terms in that sequence all seven of the first terms I'll have 10,922. So remembering that we have this really this is just the work so we have this nice little formula here let's look at this we need a sub one well that's when k equal one remember we talked about different variables could be down here we're used to i but it could be a k could be any variable so two to the first is just going to be two and the common ratio well the common ratio is the base on our exponent so the common ratio here is going to be two so if we want to find ten terms then that means the n is equal to ten s sub ten is going to be equal to a sub one which is two times one minus my r which is two but to the tenth and then we're going to divide that by one minus my rate which is two so on the bottom one minus two is going to be negative one and I'm going to go to my calculator because I'm not sure what two to the ten is so two parentheses one minus my two here at ten and I get negative 2046 and divided by negative one would be a positive 2046 so summing up the first ten terms of that formula two to the k would be 2046 now we need to talk about infinite geometric series those were all finite because we wanted to find up to a certain term but we can look at this infinite geometric series and we have this one half to the end and I want to put that in my calculator I'm going to do some exploring here so 0.5 k and then I want to look at my table so I want to look at my table and I can see that my numbers are getting smaller and smaller as I go through my table and it keeps getting farther and farther to decimal place is five now in front of that six so I keep getting more and more decimal places and eventually even if we only rounded to two decimal places this number this 17 right here would be 0.00000576 well that rounds to approximately zero so we want to do the sum of infinity infinite number so I have my ace of one one minus r to the end over one minus r but now this is going to go to infinity so this n is actually infinity well I've just said n increases this tens toward it tens zero so this would actually be then ace of one times one minus zero over one minus r oh what's one minus zero it's just one so we can say that summing up the infinite series it's just going to be ace of one times one or just ace of one over one minus r we have now found this nice little g infinite series as long as r isn't one has to be less than one and if r is greater than one then we don't actually have an infinite you can't find an infinite sum so we're looking at this one and it says determine whether it has a finite sum and if it does find the limiting value that's what does it tend toward remember it's s of infinity is equal to ace of one over one minus r ace of one is equal to ten and let's figure out what our r is it looks like it might be one half negative one half but let's double check negative five divided by ten sure enough that's negative one half and just to double check it if I have five halves divided by negative five remember you can multiply by the reciprocal so negative one over five now I have negative one half so I know that s of infinity for us is ten that's my ace of one divided by one minus r so I have one minus my r which is negative one half so we have ten to the one plus one half and your book is going to answer with fractions so we're going to say that this is ten divided by two over two plus one over two or three halves and again you're going to learn this one really well aren't you you can multiply by the reciprocal instead of dividing and that will give us twenty over three so s of infinity is equal to twenty over three and this sum keeps getting closer and closer to twenty over three that's the limiting value