1520 11 2 3 Geometric Series Source

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

for the sequence of partial sums

is called a geometric series.

You might remember learning
about geometric sequences

in the previous section.

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

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

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

which was the first term,

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

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

But the idea here is, in some cases,

we can find a formula
for the nth partial sum,

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

we can determine whether
this series will converge or diverge.

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

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

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

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

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

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

That would be n times ‘a’ [na].

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

that limit is going to be infinite.

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

Hopefully, that makes sense;

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

your sum is not going
to approach a value.

Basically, the only way
this wouldn't diverge

is if ‘a’ is equal to 0,

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

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

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

In this case, the sum of the first n terms

would just be all of these
terms added together.

So this is our nth term.

But actually, we can do something
interesting algebraically here,

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

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

I'm going to get ‘a’ times r.

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

I'm going to continue in this fashion,

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

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

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

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

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

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

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

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

But the interesting thing that happens is,

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

almost all of the terms
are going to cancel out.

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

and the very last term of this one.

Since we're solving for Sn,

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

There's also a common factor of ‘a’

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

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

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

is because we're working
under the assumption

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

What we're looking at here

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

as long as our common ratio is not 1.

What I'm interested in, though,

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

Since we have a formula for Sn,

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

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

One thing you'll maybe notice here is,

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

and it's this part right here.

This is actually something
we talked about in Section 11.1.

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

Think about it: If this is greater than 1,

then multiplying it
by itself multiple times

will cause it to basically
blow up to infinity.

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

except the sign will be alternating
from positive to negative.

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

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

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

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

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

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

We've already excluded this case,

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

It would still diverge.

What does it converge to?

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

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

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

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

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

If the absolute value of r is less than 1,

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

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

If this condition is not met,

then the geometric series
is going to diverge.