Visualizing Fourier expansion of square wave

- [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.