Proof of the Euler's Identity e^(i*pi)+1=0 by using Taylor series of e^x, sin(x) and cos(x) for calculus 2 students!
dear calculus two students after the power series section i can finally show you guys how we can get
to the most beautiful equation in math and yes that's the one i have on my shirt and this is
called the Euler's identity so have a look right here we are going to start with the power series
of e^x but instead of using x we'll be using z so let's take a look at e^z and let's look
at its expanded version this right here is equal to one plus z and the next one is plus one over
two factorial and then we have z squared and plus one over two factorial z to the third
and the beauty for this one is that we can plug in anything that we want so this is what we're going
to do it we are going to let z equal i theta and this is the reason why we use z instead of
x because z is actually a complex variable so this right here looks like we have e to the i
theta this right here is going to be 1 plus i theta just plugging i think to all the z's yeah
before we continue let's review the powers of i first one i to the first power is just equal to i
next one i squared by definition this is equal to negative 1
next i to the third power it means that we just take this and multiply by i negative 1 times i
we get negative i next i to the first power we take this and multiply by i negative i squared
which is just going to give us 1 and then i to the 5th power we just take 1 and multiply by
i and in fact you see that they are going to repeat right like this all right so in fact
if you look at this right here half of them have i half of them they don't this is just one this is i
theta and this right here i squared is actually just negative one so let me put this down as
negative and we just have theta squared okay this right here i to the third power will get negative
i yeah and then theta to the third power so it's not like alternating for now right this
is positive positive and negative negative okay and then the next one we have one so this is just
theta to the fourth power and the next one is i to the fifth power which is just plus the i and
then we have theta to the fifth power and the next one is actually going to be a negative all right
cool now i am going to put down all the terms that they don't have i first so we see
this does not have i so let's write that down and then next we have this term they don't have i
so let's put down negative and then we have 1 over 2 factorial and then we have theta squared okay
this one has i so i'm not going to put that down but this one it does not and this is a positive
so i'm going to add 1 over 4 factorial and then this is i to this is theta to the fourth power
and then in fact the next one is going to be this and it actually is a negative right
so we will just put on negative and then dot dot dot so this is the first part
now let's put down the term study of the i so starting with this and i'm actually going to
factor out that i first so let me just put down plus and here we have the i so this is just going
to be theta and then next we have this which is going to be a negative and then we have the
i which is factor there already so just have 1 over 3 factorial and then theta to the third power
and then next it's going to be this one it's positive and then we have 1 over 5 factorial
theta to the fifth power and then the next one is going to be this one here it's actually a minus
all right now take a look right here does this remind you of any power series
this right here in fact is equal to cosine theta you see it's alternating and remember
that i told you guys in my video on how you can remember power series cosine is even so you see
that you should have a alternating series and also you should see the even factorials
and also the event powers so this right here is cosine theta and then we add here we have the i
if you look at this parenthesis we see this is alternating and we have the r numbers and this
right here yes is exactly sine theta so ladies and gentlemen e to the i theta because if you are not
you get cosine theta plus i sine theta this is super super cool so let me put box and then shady
in this is just a style of like how you you are telling people that you are done with the proof
if you think calculus is just so interesting then i would highly recommend you to check our sponsor
today brilliant brilliant it's online learning platform with a big focus on interactive learning
this is from the calculus international course this course uses visual and physical intuition
to present the major topics of calculus including limits derivatives integrals and infinite series
you will walk away with a clear sense of what calculus is and what you can do all their courses
are really fun because they will always keep you engaged by storytelling and beautiful animations
besides calculus they also have a lot of other classes for you to choose from including number
theory district math probabilities and a lot more so go ahead and take a look and use the link in
the description blend downward slash black pepper pen then you can receive a 20 of discount i want
to thank blame for sponsoring this video and i also want to thank you for checking them out
so now this is what we have e to the ice data is equal to cosine theta plus i sine theta and you
see we are going to plug in pi into theta so let's put that down we let theta equal to pi and this
is the reason why with theta because it represents angle and we will get e to the i pi being equal to
cosine pi and then we add it with i times sine pi have a look on the left hand side we have e
to the i pi and then this is equal to cosine pi which is going to be negative one
well sine pi is just equal to zero zero times i doesn't matter anymore have a look
e and pi are both irrational i is imaginary and when you combine them in this fashion you end
up with a real number negative one but you know what we are not done yet because we are going
to move the negative one to the left-hand side so ladies and gentlemen here we get e to the i
pi power and then we are going to add one to it and o in o this right here is equal to zero
have a look oh my goodness this is just an equation but it has all five mathematical
constants that are considered super important namely 0 1 e and i and pi
one equation relates all five most important mathematical constants
how cool is that oh my goodness ladies and gentlemen euler's identity