Geometric series intro

let's say that I have a geometric series only a geometric sequence I should say we'll talk about series in a second so a geometric series let's say it starts at one and then our common ratio is 1/2 so the common ratio is the number that we keep multiplying by so 1 times 1/2 is 1/2 1/2 times 1/2 is 1/4 1/4 times 1/2 is 1/8 and we can keep going on and on and on forever this is an infinite geometric sequence and we can denote this we can say that this is equal to the sequence of a sub n from N equals 1 to infinity with with a sub n equaling equaling 1 times our common ratio to the N minus 1 so it's going to be our first term which is just 1 times our common ratio which is 1/2 1/2 to the N minus 1 and you can verify this right over here you can view is 1/2 to the 0 power this is 1/2 to the first power this is 1/2 squared or 1/2 to the first this is 1/2 squared so the first term is 1/2 to the 0 the second term is 1/2 to the 1 the third term is 1/2 squared so the nth term is going to be 1/2 to the n minus 1 so this is just really 1/2 to the n minus 1 power fair enough now let's say we don't just care about looking at the sequence we actually care about the sum of the sequence so we actually care about not just looking at each of these terms see what happens if I keep multiplying by 1/2 but I actually care about summing 1 plus 1/2 plus 1/4 plus 1/8 and keep going on and on and on forever so this we would now call a geometric series and because I keep adding an infinite number of terms this is an infinite geometric series so this right over here would be the infinite geometric geometric series a series you can just view as the sum of a sequence now how would we denote this well we can use summing notation we could say that this is equal to the sum we could say that this is equal to the sum let me make sure I'm not falling off the page go over to the left of route the sum from N equals one to infinity of a sub N and a sub n is just one half to the N minus one one half to the N minus one power so you just say okay what N equals one it's 1/2 to the 0 which is 1 then I'm going to sum that to when N equals 2 which is 1/2 when N equals 3 it's 1/4 on and on and on and on so all I want to do in this video is to really clarify differences between sequences and series and make you a little bit comfortable with the notation in the next few videos we'll actually try to take sums of geometric series and see if we actually get a finite value