Where did the exponential series come from?
1 + x + x^2/2! + x^3/3!...
Where did that number come from?
and how is it used to get the trigonometric series?
From Taylor (Maclaurin series if starting at x =0) series
f(x)=f(0) + f'(0)(x/1!) + f""(0)x^2/2!)....
here f(x) = e^x
f(0) = 1
the derivatives of e^x are all e^x which is 1 at x = 0
so
e^x = 1 + 1(x/1!) +1(x^2/2! etc
note
sin x = (e^ix - e^-ix)/2
cos x = (e^ix + e^-ix)/2
sin x = (e^ix - e^-ix)/2i
forgot i
cos x = (e^ix + e^-ix)/2
The exponential series, also known as the Maclaurin series for the exponential function or the Taylor series for the exponential function, was derived using Taylor series expansion. It is a mathematical representation of the exponential function, which is a fundamental mathematical function that describes exponential growth or decay.
To explain how the exponential series is derived, we start with the Taylor series expansion. The Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point. The general formula for the Taylor series expansion of a function f(x) around a point a is:
f(x) = f(a) + f'(a)(x - a)/1! + f''(a)(x - a)^2/2! + f'''(a)(x - a)^3/3! + ...
In the case of the exponential function, the derivatives of the function are straightforward. The exponential function is its own derivative, meaning that the derivative of e^x with respect to x is simply e^x. Therefore, when the expansion is centered around a = 0, also known as the Maclaurin series, the terms of the series simplify significantly.
The Taylor series expansion for the exponential function is derived as follows:
f(x) = e^x = f(0) + f'(0)x/1! + f''(0)x^2/2! + f'''(0)x^3/3! + ...
= 1 + (x^1/1!) + (x^2/2!) + (x^3/3!) + ...
This is the exponential series you mentioned: 1 + x + x^2/2! + x^3/3! ...
Now, regarding its connection to the trigonometric series, particularly the cosine and sine functions, it is related through Euler's formula. Euler's formula states that:
e^(ix) = cos(x) + i*sin(x)
Where i is the imaginary unit.
By substituting x = pi into Euler's formula, we get:
e^(i*pi) = cos(pi) + i*sin(pi)
= -1 + i*0
= -1
Using the exponential series, we can substitute x = i*pi in the series expansion for e^x:
e^(i*pi) = 1 + (i*pi)^1/1! + (i*pi)^2/2! + (i*pi)^3/3! + ...
Simplifying and separating real and imaginary terms, we find:
-1 = 1 - (pi^2/2!) + (pi^4/4!) - (pi^6/6!) + ...
0 = - (pi^2/2!) + (pi^4/4!) - (pi^6/6!) + ...
These real and imaginary terms represent the series expansions for the cosine and sine functions, respectively. The cosine series is an even function, while the sine series is an odd function. Therefore, we can use the exponential series to express the cosine and sine functions as infinite sums of powers of x.