Geometric series

In the last video we saw
that a geometric progression,

or a geometric sequence,
is just a sequence where

each successive term is the
previous term multiplied

by a fixed value.

And we call that fixed
value the common ratio.

So, for example, in this
sequence right over here,

each term is the previous
term multiplied by 2.

So 2 is our common ratio.

And any non-zero value
can be our common ratio.

It can even be a negative value.

So, for example, you could
have a geometric sequence

that looks like this.

Maybe start at one, and
maybe our common ratio,

let's say it's negative 3.

So 1 times negative
3 is negative 3.

Negative 3 times
negative 3 is positive 9.

Positive 9 times negative
3 is negative 27.

And then negative 27 times
negative 3 is positive 81.

And you could keep
going on and on and on.

What I now want to
focus on in this video

is the sum of a
geometric progression

or a geometric sequence,
and we would call that

a geometric series.

Let's scroll down a little bit.

So now we're going to talk
about geometric series, which

is really just the sum
of a geometric sequence.

So, for example,
a geometric series

would just be a sum
of this sequence.

So if we just said
1 plus negative 3,

plus 9, plus
negative 27, plus 81,

and we were to go
on, and on, and on,

this would be a
geometric series.

And we could do it
with this one up here

just to really make it
clear of what we're doing.

So if we said 3 plus 6,
plus 12, plus 24, plus 48,

this once again is a
geometric series, just

the sum of a geometric sequence
or a geometric progression.

So how would we represent this
in general terms and maybe

using sigma notation?

Well, we'll start with
whatever our first term is.

And over here if we want
to speak in general terms

we could call that
a, our first term.

So we'll start with our
first term, a, and then

each successive term
that we're going to add

is going to be a times
our common ratio.

And we'll call that
common ratio r.

So the second term is a times r.

Then the third term,
we're just going

to multiply this one times r.

So it's going to be
a times r squared.

And then we can keep going, plus
a times r to the third power.

And let's say we're going to
do a finite geometric series.

So we're not going to just
keep on going forever.

Let's say we keep
going all the way

until we get to some
a times r to the n.

a times r to the n-th power.

So how can we represent
this with sigma notation?

And I encourage you to pause the
video and try it on your own.

Well, we could think
about it this way.

And I'll give you a little hint.

You could view this term
right over here as a times r

to the 0.

And let me write it down.

This is a times r to the 0.

This is a times r to
the first, r squared,

r third, and now the pattern
might be emerging for you.

So we can write this as
the sum, so capital sigma

right over here.

We can start our index at 0.

So we could say from k
equals 0 all the way to k

equals n of a times
r to the k-th power.

And so this is,
using sigma notation,

a general way to represent
a geometric series where

r is some non-zero common ratio.

It can even be a negative value.