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# Convergent & divergent geometric series (with manipulation)

Video transcript

We have three different
geometric series here. And what I want
you to think about is, which of these converge
and which of these diverge? Or another way of
thinking about it, it converges if the sum
comes up to a finite value. It diverges if the sum
goes to infinity, or even possibly negative infinity. So pause the video
and think about that. Well, the general way to think
about it-- and we've seen it in a previous video--
is that you're going to have a converging
geometric series. Infinite geometric series
is if the absolute value of your common ratio is greater
than zero and less than one. So we just have to
think about, what is the absolute value of
the common ratios over here? And it's not as obvious,
because they didn't just write one term or one
number to an exponent. They wrote several
numbers to an exponents. So we'll have to do a little
bit of algebraic manipulation. So 5 n to the negative
1, that's the same thing. Let me do that over here. 5 n to the negative 1
is the same thing as 5 to the n over 5. Or you could write it
as 5 to the n times 5 to the negative 1, which
is the same thing as 5 to the n over 5. And so 5 to the n over 5, 5
to the n over 5 times 9 10 to the n. 9 over 10 to the n. This is the same thing. This is equal to 1 over
5 times 5 times 9 over 10 to the nth power. And now the common ratio
becomes a little bit more clear. This right over here
would be our-- well, it won't be our
first term anymore, because we're starting
at n equals 2. So we won't view it that way. But if we look at our common
ratio, this right over here is going to be our common ratio. And this is absolute value
greater than or less than 1. Well, 5 times 9 is
45, divided by 10. That's going to
give you 4 and 1/2. So this is greater than 1. The absolute value
is greater than 1. So this one is going to diverge. This sum is going
to go to infinity. Now let's think about
this one over here. I encourage you to
pause it if that one helped you a little bit. Let's try to rewrite
this in a way that the common ratio is
a little bit more obvious. So 3/2 to the n times 1
over 9 to the n plus 2. Well, we could think
about this as 1-- let me write it this way. 3/2 to the n times 1
over 9 to the n plus 2 is the same thing as-- let me
write it this way-- 9 to the n times 9 squared. And if we write
it that way, then this is going to be the
same thing as-- let's write it this way-- 1 over 9
squared, which is 81, times 3/2 to the n times 1 over 9 to the
n is the same thing as 1 over 9 to the n. Notice, 1 over 9 to the n,
if I raise 1 to the n power, it's not going to change
the value of that 1. And so that's going to be the
same thing as 1/81 times-- let's see. 3 divided by 9, we're going
to get 1 over 6 to the n. And so here it's a
little bit clearer that our common ratio is 1/6. Its absolute value is
clearly less than 1. So this is going to converge. This is actually going to give
you a finite value for the sum. Finally, let's go on to
this one right over here. So let's see. 1 over 3 to the n minus 1. So let me rewrite
the whole thing. This is 2 to the n. And I'm going to write it
over 3 to the n times 3 to the negative 1. Same exact thing, which
is the same thing as 3 to the negative 1
in the denominator. I could write it
in the numerator. So it's going to be 3
times 2 to the n over 3 to the n, which is equal
to 3 times 2/3 to the n. So all this business is the same
thing as 3 times 2/3 to the n. Common ratio-- its
absolute value of 2/3-- is clearly less than 1. So once again, this infinite
geometric series will converge. It will give you a finite value.