Maclurin for sin(x)
f(x) = sin(x)
f'(x) = cos(x)
f''(x) = -sin(x)
f'''(x) = -cos(x)
f''''(x) = sin(x)
And this repeats forever.
f(0) = 0
f'(0) = 1
f''(0)=0
f'''(0)=-1
f''''(0)=0
And this repeats forever.
From here we can center a maclurin series at (a)? = 0 -> I cannot remember if its a or x
Using 5 points we can use
f(0) = 0
f'(0) = 1
f'''(0) = -1
f'''''(0) = 1
f'''''''(0) = -1
From here we can conclude (1) we are oscillating between - and +, (2) the points are always odd multiples of primes, and (3) since the even multiples of prime are 0, we can ignore these as they do not add value
The defintion of odd is defined as 2k+1, k element of the real
So
(-1)^(n)(x^(2k+1))/(2k+1)!
And we can add this at every n.
If we add this add every n we can think of an infinite sum
Since we are centered at (a?) = 0 the sum must start at 0.
To motivate convergence, we plug in known points of sin(x).
Must it be that if our sum at some n equals the function at the same x,
Out infinite sum converges for all x?
Let me know if i did justice for the maclurin of sin(x)!
Sorry posted via phone.... don't know how to use symbols here
