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One type of series where we can 
actually come up with a formula

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for the sequence of partial sums

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is called a geometric series.

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You might remember learning 
about geometric sequences

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in the previous section.

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Basically, it's the same idea except 
the terms are being added together.

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Every term is being multiplied 
by a common ratio, which we call r,

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and that means the formula 
for our generic nth term would be ‘a’,

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which was the first term,

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and then times r to the 
n minus 1st power,

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so that would be the formula for 
our series for each individual term.

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But the idea here is, in some cases,

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we can find a formula 
for the nth partial sum,

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for the sum of the first n terms, 
and based on that,

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we can determine whether 
this series will converge or diverge.

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A couple of cases to get 
out of the way first,

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if r is equal to 1, 
what would this thing look like?

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Well, what it would look like is 
just ‘a’ and then plus another ‘a’

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and then plus another ‘a’ and 
continuing in that fashion forever.

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Now, if I just added up 
the first n terms here,

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I would have a total of n a’s 
all being added together.

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That would be n times ‘a’ [na].

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But if we take a limit as n 
approaches infinity for n times ‘a’,

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that limit is going to be infinite.

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So if r is equal to 1, 
this series is going to diverge.

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Hopefully, that makes sense;

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if you add up an infinite number of 
terms and they're all staying the same,

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your sum is not going 
to approach a value.

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Basically, the only way 
this wouldn't diverge

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is if ‘a’ is equal to 0,

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and most of the time, 
we're going to ignore that case

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because it's really not a very interesting 
version of a geometric series.

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Next up, we're going to look 
at the case: if r is not equal to 1.

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In this case, the sum of the first n terms

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would just be all of these 
terms added together.

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So this is our nth term.

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But actually, we can do something 
interesting algebraically here

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if directly below this, 
I take r times all of the terms,

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because if I do that, 
I take r times this term,

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I'm going to get ‘a’ times r.

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If I take r times this term, 
I'll get ‘a’ times r squared.

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I'm going to continue in this fashion,

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and this is actually going to 
be the second-to-last term

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because that's what I would get 
if I took the previous term times r.

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That means, if I take this times r, 
I'm going to get ‘a’ times r to the n.

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And it turns out, since both 
of these sums are finite,

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(there's a set number of terms),
I can combine them together.

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But surprisingly, 
I'm actually going to subtract them.

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I'm going to get Sn minus r times Sn.

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My goal here is to come up 
with a formula for just Sn.

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But the interesting thing that happens is,

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if we were to subtract all of 
these terms from these terms,

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almost all of the terms 
are going to cancel out.

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The only ones that are going 
to be left are the very first term

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and the very last term of this one.

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Since we're solving for Sn,

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it hopefully makes sense to factor 
that out from this side of the equation.

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There's also a common factor of ‘a’

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that we could factor out 
from this side of the equation.

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Solving for Sn, we can get this 
by itself by dividing by 1 minus r.

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The reason we don't have 
to worry about dividing by 0

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is because we're working 
under the assumption

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that r is not equal to 1 in this case, 
which gives us this formula.

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What we're looking at here

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is a formula for the sum of the first 
n terms of a geometric series

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as long as our common ratio is not 1.

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What I'm interested in, though,

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is if the infinite series is 
going to converge or not.

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Since we have a formula for Sn,

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now we're going to take the limit 
of this as n approaches infinity

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to see what's going to happen to the sum.

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One thing you'll maybe notice here is,

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there's only one part of this
entire formula that has an n in it,

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and it's this part right here.

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This is actually something 
we talked about in Section 11.1.

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This right here is only going to converge 
if the absolute value of r is less than 1.

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Think about it: If this is greater than 1,

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then multiplying it 
by itself multiple times

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will cause it to basically 
blow up to infinity.

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And if it's less than negative 1, 
the same problem is going to happen,

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except the sign will be alternating 
from positive to negative.

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This is only going to converge 
if the absolute value of r is less than 1.

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Any other value for r, 
and this thing is going to diverge,

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except of course, 1, 
which we've already ruled out.

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Basically, this sequence, which stands
for the sequence of partial sums,

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converges if the absolute 
value of r is less than 1,

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and it's going to diverge if the absolute 
value of r is greater than or equal to 1.

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We've already excluded this case,

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but if r was negative 1, 
we would run into the same problem.

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It would still diverge.

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What does it converge to?

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Well, if the absolute 
value of r is less than 1,

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then this term right here is 
going to approach 0 as n grows.

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And that means, 
what we're going to be left with

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is just ‘a’ in the numerator 
divided by 1 minus r,

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and that winds up being 
the sum of a geometric series.

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If the absolute value of r is less than 1,

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then this series (the geometric series) 
is going to have a sum,

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and that sum is going to equal 
‘a’ divided by 1 minus r.

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If this condition is not met,

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then the geometric series 
is going to diverge.