Taylors Theorem

From Mean Value Theorem, a function f(a+b) can be written as

f(a+b)=f(a)+bf(c)

where c is some point between a and a + b. By writing the definition of c in this way, we have a statement that works whether b > 0 or b < 0. We have already met the approximation

f(a+b)~f(a)++bf(c)

when we studied the Newton - Raphson method for solving an equation, and have already observed that the Mean Value Theorem provides a more accurate version of this. Now consider what happens when f is a polynomial of degree n

f(x)=a_0+a_1x+a_2x^2+a_3x^3+\cdot \cdot \cdot +a_{n-1}x^{n-1}+a_{n}x^{n}

Note that f(0) = a_0. Differentiating this equation gives us

f'(x)=a_1+2a_2x+\cdot \cdot \cdot +(n-1)a_{n-1}x^{n-2}+na_{n}x^{n-1}

and so f(0) = a_1. Differentiating this again would give us,

f''(x)=2a_2+6a_3x + \cdot \cdot \cdot +(n-2)(n-1)a_{n-1}x^{n-3}+n(n-1)a_{n}x^{n-2}

Therefore, our f''(0) = 2a_2 . After the next differentiation, we get f''(0) = 3!a_3. We can generalize this behavior as f^k(0) = k!a_k. Thus we can rewrite the polynomial, using its value, and the value of its derivatives at 0, as

f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\frac{f'''(0)}{3!}x^3+\cdot \cdot \cdot +\frac{f^{n-1}(0)}{\left (n-1 \right )!}x^{n-1}+\frac{f^{n}(0)}{n!}x^n

This opens up the possibility of representing more general functions than polynomials in this way, and so getting a generalization of the Mean Value Theorem. This is now the fundamentals of Taylor and Maclaurin Series and is called Taylor’s Theorem.

In general, we can restate the Taylor’s theorem as

f(x)=f(c)+f'(x)(x-c)+\frac{f''(c)}{2!}\left (x-c \right )^2+\frac{f'''(c)}{3!}\left (x-c \right )^3+\cdot \cdot \cdot

for some c between a and x.

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