Original subtitles

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In this video we're going to be talking about how to find the sum of a geometric series.

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And if we want to find the sum of a geometric series we first have to be able to determine that the

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geometric series converges and in order to say that the geometric series converges of course we have

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to be able to say that we are in fact dealing with a geometric series instead of some other kind of

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series.

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So working backwards that way we're going to kind of think about this problem in three steps we've been

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given this series and we haven't been told explicitly it's a geometric series we're going to find that

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out a second.

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But what we want to do is first show that this is in fact a geometric series which will do by getting

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the series into this form here then we'll say whether or not the geometric series converges.

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And then if it does converge we'll find some using this formula.

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So first of all let's talk about how to say whether or not this is a geometric series.

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If it is a geometric series we'll be able to get it into this form here.

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The infinite sum from end equals zero to infinity of a Times are to the end power and we'll come back

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to that in a second.

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The value we've been given is this infinite or repeating decimal zero point to one with the bar over

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the two one which means that the 2 1 repeats.

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So this is we can say the same thing as zero point two one two one two one two one repeating forever

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so that to 1 just repeats.

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So we're looking to do is turn this into a geometric series.

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And what we want to recognize is that we have each of these two one pairs.

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We can turn into a fraction.

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So if we just look at this point to one here.

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We know that that's the same as twenty one over 100 so we can call that 21 over 100.

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We know it's over 100 because this number ends in the hundredths place.

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The two is in the 10th place the one is in the hundredths place.

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So since this point to one here ends in the hundredths place it's the same as 21 over 100.

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What about the second 21 here.

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This 2 one while the two is in the thousands place the one is in the 10000 place.

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So what we want to do is add to this 21 over 100 we want to add 21 over ten thousand because 21 over

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10000 represents this to one here.

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And if we add these two fractions together we get Point 2 1 to 1 to deal with the next 21 here.

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This 21 the two is going to be in the 100000 place which means the one is going to be in the millionths

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place.

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So what we want to do is go ahead and add it to this.

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Some here we want to add 21 over 1 million and noticed that we're just adding two zeros to the denominator

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every time and that should make sense because we have a number here that repeats it's a two digit number.

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So once we identify this 21 over 100 We know that the next 21 since it's two digits it's just going

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to have two more zeros here in the denominator so we add two zeroes.

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We add two zeros and we could do that forever.

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For now we're just going to say plus dot dot dot because this goes on forever and we're just looking

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for a sum that represents this repeating decimal.

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So once we have it in this form what we can do is factor out a 21 over 100 from this sum.

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So if we factor out 21 over 100 we have to multiply 21 over 100 by one to get this 21 over 100 then

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we're going to add to that here.

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What do we have to multiply by 21 over 100 to get 21 over 10000.

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Well we have to multiply the numerator by 1 21 times 1 gives us 21.

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So the numerator is going to be 1.

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What do we have to multiply by 100 to get 10000.

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Well that's 100 100 times 100 gives us 10000.

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And remember you can just add the zeros here and 10000 we have 1 2 3 4 zeros.

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That means we have to have 1 2 3 4 zeros between these two to get four zeros we multiply them together.

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So we get plus 1 over 100 and then we're going to get plus 1 over 100.

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And again we just add two more zeros.

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So one hundred times 10000 gives us 1 million and we'll go ahead and say plus dot dot dot.

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And now we factored 21 over 100 out of the sum.

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Now if we just look at the values here inside the parentheses if we keep this 21 over 100 out in front

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if we look at the values inside these parentheses here.

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Let's start with the one over 100.

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Well that's the same thing as saying one over 100 raised to the first power one over 100 raised the

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first power is just 1 over 100.

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So thinking one over 10000 is the same thing as quantity 1 over 100 squared.

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Remember anything raised to the power of 0 is just one.

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That means that we can represent one here with one over 100 raised to the zero power.

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So what you can see here then is that we're starting to get this pattern here.

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One over 100 raised to the zero power the first power the second power.

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Which means if we went on forever the next turn would be 1 over 100 raised to the third power Fourth

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Power fifth power.

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And so what's starting to emerge is a geometric series.

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Because if we compare this series to this infinite sum equals zero to infinity of a R to the N.

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But we can recognize here is that a is a coefficient that's multiplied by this art of the N series.

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Well in our case is just 21 over 100 it's a value that's multiplied out in front by this series here.

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Notice here than in this series and starts at zero which means we're looking for art of the zero plus

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the first Plus are the second.

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Well that looks awfully similar to this one over 100 0 plus 1 over 100 for the first plus 1 over 100

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to the second et cetera.

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So then what we identify is that this one over 100 value is the same as are here.

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And that because START starts zero we see and starting down here 0 1 2 and counting up.

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So essentially by getting the series into this forum we've identified a value for A and a value for

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our.

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Once you get the series to this point here you don't have to go to the next step you can identify and

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are right at this point a is always going to be this value out in front of the parentheses.

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Is always going to be this second term here.

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The turn immediately following the 1.

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So as long as the series Inside these parentheses here starts with 1 which is why we factored out the

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21 over 100 because we wanted this to start with one as long as we've gotten this to start with one

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then that means that this is going to be a.

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And this is going to be our Assuming of course that you get are the first R-squared are the third are

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the fourth etc. which in this case we do.

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So we've identified a and we've identified our.

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And we've shown that this series has the same form as the series right here which means that this is

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in fact a geometric series because it has the same form as this series.

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And we can identify n are we know that this is a geometric series.

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So a sense of the geometric series we can use this geometric series Test to say whether or not the series

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converges according to the geometric series test a geometric series will converge with the absolute

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value of r is less than 1.

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Well we know in this case that our is 1 over 100.

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We've identified that already.

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So we can say the absolute value of 1 over 100 less than 1.

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And in fact that is true there there's no negative sign here so the absolute value of 1 over 100 is

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just 1 over 100 and that is less than 1.

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So the series will in fact converge if this value one over 100 was greater than or equal to 1.

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We would know that the geometric series was divergent but because it's less than one we can say that

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this particular geometric series converges or that it's convergent.

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So because it converges we can find it some if it was divergent we wouldn't be able to find the sum

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but because it's convergent we can find some.

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So what's the sum of this geometric series.

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Well the sum is always given by this formula a divided by 1 minus r.

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And when we say the same here what we're talking about is this.

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Some over here.

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So this whole sum right here the sum from N equals zero to infinity of a part of the N.

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This whole series if we want to find the sum of this series the sum is going to be a over 1 minus or

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so sometimes you'll see the formula where you'll see this infinite sum set equal to a over 1 minus R

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because they're equal to one another.

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So in order to find the sum we're just going to plug in a and r to this a over one minus our formula

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and then we're going to simplify.

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So we know that a.

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In our case is equal to 21 over 100.

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So we're going to put that in the numerator and then in the denominator we're going to get 1 minus r

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r we know is 1 over 100 so we're going to get 1 over 100.

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And now we want to go ahead and simplify this.

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And you're always going to simplify in the same way.

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So instead of one here in the denominator we're going to find a common denominator with this fractions

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instead of one we're going to get 100 over 100 that's the same thing as 1.

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That way we can combine these two fractions in the denominator and say 100 minus one is ninety nine.

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So this becomes 21 over 100 divided by 99 over 100.

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And I remember that when we have a fraction divided by a fraction we keep the fraction in the numerator.

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21 over 100.

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And instead of dividing by the fraction from the denominator we multiply by it's reciprocal.

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So instead of division we'll do multiplication and the reciprocal is whatever we get when we flip this

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fraction upside down.

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So instead of 99 over 100 we'll say 100 over ninety nine and we haven't changed anything.

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This multiplication is the same thing as this division here.

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Now though with the multiplication we can clearly see that 100 is going to cancel with 100 We have 1

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in the denominator and 1 in the numerator and then we can say that our final answer the sum of the series

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is going to be 21 over ninety nine and we can leave it in fraction form.

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Or we could use our calculators to find a decimal representation of 21 over 99 but 21 over 99 is going

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to be the sum of this convergent geometric series.