If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content
Current time:0:00Total duration:4:46

Proof of infinite geometric series as a limit

Video transcript

in a previous video we derived the formula for the sum of a finite geometric series where a is the first term and R is our common ratio what I want to do in this video is now think about the sum of an infinite geometric series and I've always find this found this mildly mildly mind-blowing because we're actually more than mildly my mind blowing because you're taking the sum of an infinite things but as we see you can actually get a finite value depending on what your common ratio is so there's a couple of ways to think about it one is you could say that this the sum of an infinite geometric series is just the limit of this as n approaches infinity so we could say we could say what is the limit what is the limit as n approaches infinity as n approaches infinity of this business of the sum from k equals 0 from k equals 0 to n of a times R to the K which would be the same thing as taking the limit as n approaches infinity right over here so that would be the same thing as the limit as n approaches infinity of all of this business let me just copy and paste that so I don't have to keep switching colors so copy and then paste so what's the limit as n approaches infinity here let's think about that for a second I encourage you to pause the video and I'll give you one hint think about it for ARS greater than 1 for R is equal to 1 and actually let me make it clear let's think about it for the absolute values of R is greater than 1 the absolute values of R equal to 1 and then the absolute value of R less than 1 well I'm assuming you've given a go at it so if R if the absolute value of R is greater than 1 as this exponent explodes as it as it approaches infinity this number is just going to become massively massively huge and so the whole thing is just going to become or at least you could think of the absolute value of the whole thing is just going to become a very very very large number if R was equal to 1 then the denominator is going to become zero and we're going to be dividing by that denominator and this formula just breaks down but where this formula can be helpful and where we can get this to actually give us a sensible result is when the absolute value of R is between 0 & 1 we've already talked about we're not even dealing with the geometric we're not even talking about a geometric series if R is equal to 0 so let's think about the case let's think about the case where the absolute value of R is greater than 0 and it is less than 1 what's going to happen in that case well the denominator is going to make sense right over here and then up here what's going to happen well if you take if you take something with an absolute value less than 1 and you take it to higher and higher and higher exponents every time you multiply it by itself you're going to get a number with a smaller absolute value so this term right over here this entire term is going to go to 0 as n approaches infinity you imagine if R was 1/2 you're talking about 1/2 to the hundredth power 1/2 to the thousandth power 1/2 to the millionth power what happened the billionth power that quickly approaches zero so this goes to zero if the absolute value of R is less than 1 so this we could argue would be equal to would be equal to a over 1 minus R a over 1 minus R so for example if I had the geometric series if I had the infinite geometric series let's just have a simple one let's say that my first term is 1 and then each successive term I'm going to multiply by 1/3 so it's 1 plus 1/3 plus 1/3 squared plus 1/3 to the third plus and I were to just keep on going forever this is telling us that that sum this infinite sum have an infinite number of terms here this is this is a pretty fascinating concept here we'll come out to this it's going to be my first term it's going to be my first term 1 over one - my common ratio one - my common ratio in this case is 1 third one minus one third which is the same thing as one over two thirds which is equal to three halves or you could view it as one and a half that's a mildly amazing thing