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Today we're going to be talking about how to determine whether or not a geometric series is convergent

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or divergent.

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In other words we're going to be using the geometric series test for convergence and the series.

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In this particular video that we're going to be testing is the infinite sum from Ed equals one to infinity

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of this series here which is negative 3 Race to the end minus one power divided by 4 to the power.

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Now as a reminder I have the geometric series convergence test written here or at least part of it.

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It says that the infinite sum from Ed equals one to infinity.

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Of this type of series which is a geometric series a times are raised to the end minus one power that

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that series is convergent.

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If the absolute value of r is less than 1 and divergent if the absolute value of r is greater than or

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equal to 1.

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Now the easiest way to use this geometric series convergence test is to first expand this series here

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that were given in the definition into a series.

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And the way that we're going to do that is by plugging in values of and starting with N equals 1.

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We'll keep plugging in higher and higher values of and and just get the first couple of terms of this

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series then it'll become much more apparent to us what kind of series a geometric series actually is.

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So we'll go ahead and just say that this is going to be roughly equal to here.

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When we plug in and equals 1 to this series here we at a times are to the one minus one power one minus

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one is ZERO are raised to the zero power is 1.

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So we essentially have here a times one and we just get a.

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Ok then we're going to add to that whatever we get when we plug in and equal to 2 minus 1 gives us once

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we get a times are to the first power or just a r.

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So plus a R when we plug in unequals 3 we get 3 minus 1 which is 2 or a r squared.

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So plus a r squared.

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And if we kept going here what we'd see is that we get a r cubed a r to the fourth etc..

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I mean just going to say Dot dot dot here.

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OK.

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So this is the expanded form here of a geometric series.

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Now what we can do what's really interesting here is that we can factor out the value from this series.

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Notice that each term in the series is multiplied by a.

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So when we factor out a like this what we get is one right a times one is just a that gives us our first

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term plus our when we factor in the second term a times are gives us a r here and we keep factoring

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out we just get plus R squared plus R Q plus dot dot dot.

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We get to the fourth R the fifth etc..

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So here's our geometric series now with a factored out.

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The reason it's convenient to understand that this is the expanded form of a geometric series is because

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we can now plug in values of an AND equals 1 2 3 etc. into our series appear to see if it matches this

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form.

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A Times 1 plus R plus R-squared plus argued to see if it matches that form.

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If it does match that form then we can tell that it's a geometric series.

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We can easily tell it's a geometric series and we can also easily identify values of a and r which is

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what we're interested in not only for the convergence test but also for finding the sum of a geometric

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series.

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If we need to do that as well.

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So again the easiest way to determine whether or not we actually have a geometric series here is just

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a plug in the first several terms.

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So let's go ahead and do that.

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Here are series and we'll just go ahead and put it in brackets.

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When we plug and equals one into this series here we'll get in the numerator one minus one which is

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zero.

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Negative 3 raise to the zero power as just once we get 1 in the numerator in the denominator when we

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plug in one we get four to the first power which is just 4.

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So our first term in our series is one fourth that we're going to add to that whatever we get only plug

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in 2.

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So I plug in to we get to minus 1 which is 1 negative 3 raise to the first power is just negative 3

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so negative 3 divided by four raised to the second power of four squared here which is 16.

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So we get negative 3 over 16 for our second term.

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Now let's just do two more really quickly.

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When we plug in three we get three minus one which is to negative three squared is nine.

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So we get positive nine divided by four cubes which is 64.

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So nine over 64.

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And if we plug in an equals four we get four minus one which has three negative three cubed is negative

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27.

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So negative 27 divided by four to the fourth power which is two hundred fifty six and we could just

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keep going but we'll go ahead and say dot dot dot like this and close our series.

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OK.

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So these are the first several terms of our series.

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Now in order to determine whether or not this is a geometric series once you've got these terms right

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now what you always want to try to do is factor out the value of our first term.

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Remember here we had our first term a and we tried to factor out a.

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So in our case we have our first term one fourth and we want to try to factor out one fourth.

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So what we're going to do is say one fourth and then multiply this by all these terms in the inside

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here.

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So one fourth obviously the first term will be multiplied by 1 to get one fourth again.

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Now what do we have to multiply by one fourth in order to get negative 3 over 16.

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Well we have to multiply that by negative 3 over 4 right now if we take a negative three fourths and

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we multiply it by one fourth.

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We get negative three in the numerator and 16 in the denominator which is what we have here for our

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second term negative 3 over 16 where we have to multiply by one fourth in order to get nine over 64.

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Well we have to multiply that by nine.

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And then in the denominator we have to multiply that by 16 and we could keep going here we have to multiply

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this by negative twenty seven over 60.

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For me it's a plus dot dot dot.

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And this would be our series with the first term.

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One fourth factor it out.

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Now all we can do to verify that this is a geometric series is make sure that we have it in the same

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form as our series here and identify a and are.

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Well theoretically if this is a geometric series we already factored out our value of a we're already

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assuming that a is one fourth because it was our first term we got we went ahead and factor that out.

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Now our is always going to be our second term here and we want to make sure that we always include a

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negative sign if there is one.

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So this is going to be our value of our including the sign in front of it whether it's positive or negative.

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So in our case that would be negative three fourths would be our value of our.

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Now notice that all we have to do is make sure that the terms following are are in fact are squared

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are cubed art of the fourth et cetera.

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So what we want to do is go ahead and break these down to see if we in fact say that if this is our

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then this is our squared.

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This is our cubes et cetera.

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So we'll have one fourth multiplied by one minus three fourths.

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And now what we're going to try to do is get three fourths squared.

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And obviously we can see that that is going to be the case we're going to say plus three fourths squared

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here because we can see that if we square three we get nine if we square four we get 16.

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That's nine sixteenths which was r r squared term.

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And in fact what we can do is because we know that our value for our is negative three fourths we can

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go ahead and add in our negative rate here.

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Negative three four square gives a positive nine 16th which is what we need here for our fourth term

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this 27 or 64 we can go ahead and say plus and then negative three fourths cubed.

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Because when we cube the negative will still end up with a negative right negative one race to the third

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power still negative one we get this negative sign out in front and three cubes is 27 and four cubed

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is sixty four.

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So we're getting that value.

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OK so we've proven we've shown that negative three fourths here.

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This second term inside this series is in fact our value of our.

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And that if we keep going with that we get R-squared argued of the fourth.

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So we've shown that this is a geometric series because we've matched the form of our series to the definition

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of a geometric series here.

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And we know that one fourth is equal to a and then negative three fourths is equal to r.

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So now when it comes to the convergence test all we need to do is look at our right we're taking the

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absolute value of our.

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So we want the absolute value of negative three fourths which of course is equal to positive three fourths.

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Is this less than or greater than or equal to 1.

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Of course it's less than 1.

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So because three fourths is less than 1 We know that the series is convergence so we'll say that it

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is a convergent series by the geometric series tests.

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If this value had been greater than or equal to one we would know that the series is divergent.

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And now as a quick note about the sum of the series once you have a and are finding the sum of the series

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is very easy.

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The sum of a geometric series is always just a divided by 1 minus R and since we already have values

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of a n r that's very easy.

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So our value of a is one fourth our value for our is negative three fourths so you get 1 minus negative

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three fourths or 1 plus three fourths.

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And now we just need to simplify this to find the sum.

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The way that you want to do this is by finding a common denominator within the denominator so this one

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here becomes for over 4 to find a common denominator with this three fourths.

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That leads us to one fourth divided by seven fourths.

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Well when we have a fraction divided by a fraction instead of dividing by a fraction we can multiply

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by its reciprocal.

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So we'll get one fourth and instead of dividing that by seven fourths we'll multiply it by the reciprocal

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four sevenths.

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When we do that we'll of course get our force here in the numerator and denominator to cancel with one

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another and we're left with a value of 1 7 which is the sum of the geometric series.

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See how quick that was to find the sum.

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Once we have values of A and are it's very easy.

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So that's how you use the geometric series test to determine whether or not a geometric series is convergent

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or divergent.

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Remember that the easiest way to determine whether or not this is in fact a geometric series is just

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to go ahead and write down the first few terms in the series and then see if you can get it into this

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form here.

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If you cannot get it into this form then it's not a geometric series what makes it a geometric series

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is this value here of R which is the constant multipole right each term.

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After this we're multiplying by another negative three fourths So we have negative three fourths to

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the first and then negative three fourths times negative three fourths negative three fourths times

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three fourths I'm saying three four so we're always just multiplying the previous term by an additional

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factor of R.

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That's what makes it a geometric series.

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If you can't get in this form it's not a geometric series.