Series as sum of sequence - Mathematics, Engineering Video Lecture - Engineering Mathematics

Let's say that I have
a geometric series.
A geometric sequence,
I should say.
We'll talk about
series in a second.
So a geometric series,
let's say it starts at 1,
and then our common
ratio is 1/2.
So the common
ratio is the number
that we keep multiplying by.
So 1 times 1/2 is 1/2, 1/2
times 1/2 is 1/4, 1/4 times 1/2
is 1/8, and we can keep going
on and on and on forever.
This is an infinite
geometric sequence.
And we can denote this.
We can say that this is equal to
the sequence of a sub n from n
equals 1 to infinity, with
a sub n equaling 1 times
our common ratio
to the n minus 1.
So it's going to be our
first term, which is just 1,
times our common
ratio, which is 1/2.
1/2 to the n minus 1.
And you can verify it.
This right over here you can
view as 1/2 to the 0 power.
This is 1/2 half to the first
power, this is 1/2 squared.
1/2 to the first,
this is 1/2 squared.
So the first term
is 1/2 to the 0.
The second term is 1/2 to the 1.
The third term is 1/2 squared.
So the nth term is going
to be 1/2 to the n minus 1.
So this is just really 1/2
to the n minus 1 power.
Fair enough.
Now, let's say we
don't just care
about looking at the sequence.
We actually care about
the sum of the sequence.
So we actually
care about not just
looking at each of
these terms, see
what happens as I keep
multiplying by 1/2,
but I actually care
about summing 1 plus 1/2
plus 1/4 plus 1/8, and keep
going on and on and on forever.
So this we would now
call a geometric series.
And because I keep adding
an infinite number of terms,
this is an infinite
geometric series.
So this right over here would be
the infinite geometric series.
A series you can just view
as the sum of a sequence.
Now, how would we denote this?
Well, we can use
summing notation.
We could say that this
is equal to the sum.
We could say that this
is equal to the sum.
Let me make sure I'm not
falling off the page.
Let me just scroll
over to the left a bit.
The sum from n equals 1
to infinity of a sub n.
And a sub n is just
1/2 to the n minus 1.
1/2 to the n minus 1 power.
So you just say OK,
when n equals 1,
it's 1/2 to the 0, which is 1.
Then I'm going to
sum that to when
n equals 2, which is 1/2,
when n equals 3, it's 1/4.
On and on, and on, and on.
So all I want to do in this
video is to really clarify
differences between
sequences and series,
and make you a little bit
comfortable with the notation.
In the next few
videos, we'll actually
try to take sums
of geometric series
and see if we actually
get a finite value.