- [Voiceover] So we started
with a square wave
that had a period of two pi,
then we said, hmm,
can we represent it
as an infinite series
of weighted sines and cosines,
and then working from that idea,
we were actually able
to find expressions
for the coefficients,
for a sub zero and a sub n
when n does not equal zero,
and the b sub ns.
And evaluating it for this
particular square wave,
we were able to get
that a sub n is going
to be equal,
or a sub zero is going
to be 3/2,
that a sub n is going
to be equal to zero
for any n
other than zero,
and that b sub n
is going to be equal
to zero if n is even
and six over n pi
if n is odd.
So one way to think about it,
you're gonna get your a sub zero,
you're not gonna have any
of the cosine terms,
and you're only going to have
the odd sine terms.
And if you think about it
just visually, if you look
at the square wave,
it makes sense
that you're gonna have
the sines and not the cosines
because a sine function
is gonna look something like this.
So a sine function is gonna look
something like this,
while a cosine function
looks something like,
let me make it a little bit neater,
a cosine function would look
something like that.
And so a cosine
and multiples of cosine
of two x, cosine of three x,
is gonna be out of phase,
while the sine of x,
or I should say cosine of ts
and the sines of ts,
sine two t, sine three t,
is gonna be more in phase
with the way this function
just happened to be.
So it made sense that our
a sub ns were all zero
for n not equaling zero.
And so based on what we found
for our a sub zero,
and our a sub ns,
and our b sub ns,
we could expand out
this actual,
we did in the previous video,
what is this Fourier series
actually look like?
So 3/2 plus six over pi
sine of t plus six over three pi
sine of three t plus
six over five pi
sine of five t,
and so on and so forth.
And so a lot of you might
be curious what does this
actually look like.
And so I actually just,
you can type these things
into Google and it will
just graph it for you.
And so this right over here
is just the first two terms.
This is 3/2 plus six over pi
sine of t.
And notice it's starting
to look right because our
square wave looks something
like, it goes,
it looks something like this.
So it's gonna go like that
and then it's gonna go down
to zero and then it's gonna go
up, looks something like that.
It doesn't have the pis
and the two pis
marked off between these
because it's gonna look
something like that.
So even just the two terms,
it's kind of a decent approximation
for even two terms,
but then as soon as you get
to three terms,
if you add the six
over three pi sine of three t
to the first two terms.
So if you look at these
first three terms,
now it's looking a lot
more like a square wave.
And then if you add the next term,
well, it looks like even more
like a square wave,
and then if you add to that
what we already wrote down here,
if you were to add to that
six over seven pi times
sine of seven t,
it looks even more
like a square wave.
So this is pretty neat.
You can visually see
that we were actually able
to do it.
And it all kind of just
fell out from the mathematics.