6.1 Taylor Theorem
\[ f^{(1)}(x) = \frac{dy}{dx}, \; f^{(2)}(x) = \frac{d^2 y}{dx^2}, \; \cdots \; , f^{(n)}(x) = \frac{d^n y}{dx^n} \]
Taylor theorem
\[ f(b) = f(a) + \frac{f^{(1)}(a)}{1!} (b - a) + \frac{f^{(2)}(a)}{2!} (b - a)^2 + \cdots + \frac{f^{(n-1)}(a)}{(n-1)!} (b - a)^{n-1} + \int_a^b \frac{(b-t)^{n-1}}{(n-1)!}f^{(n)}(t)dt \]
\[ f(b) = f(a) + \frac{f^{(1)}(a)}{1!} (b - a) + \frac{f^{(2)}(a)}{2!} (b - a)^2 + \cdots + \frac{f^{(n-1)}(a)}{(n-1)!} (b - a)^{n-1} + R_n \]
Proof. n = 1
\[ f(b) = f(a) + \int_a^b \frac{((b-t)^{1-1}}{(1-1)!} f^{(1)}(t) dt \]
\[ f(b) - f(a) = \int_a^b f^{(1)}(t) dt = R_1 \]
n = 2
\[ R_1 = \int_a^b \frac{(b-t)^{1-1}}{(1-1)!} f^{(1)}(t)dt = \int_a^b f^{(1)}(t) \frac{(b-t)^{1-1}}{(1-1)!} dt = \int udv \]
\[ u = f^{(1)}(t), \; dv = dt \] \[ du = f^{(2)}(t)dt, \; v = -(b-t) \]
\[\begin{equation*} \begin{split} R_1 &= \int_a^b f^{(1)}(t) \frac{(b-t)^{1-1}}{(1-1)!} dt = \int udv = \left. uv \right| - \int vdu \\ &= \left. -f^{(1)}(t)(b-t) \right|_a^b -\int_a^b -(b-t)f^{(2)}(t)dt \\ &= f^{(1)}(a)(b-a) + \int_a^b (b-t)f^{(2)}(t)dt \\ &= f^{(1)}(a)(b-a) + R_2 \end{split} \end{equation*}\]
\[ f(b) - f(a) = \int_a^b f^{(1)}(t) dt = R_1 = f^{(1)}(t)(b-a) + R_2 \]
n = 3
\[ R_2 = \int_a^b (b-t) f^{(2)}(t)dt = \int_a^b f^{(2)}(t) (b-t) dt = \int udv \] \[ u = f^{(2)}(t), \; dv = (b-t)dt \] \[ du = f^{(3)}(t)dt, \; v = -\frac{(b-t)^2}{2} \]
\[\begin{equation*} \begin{split} R_2 &= \int_a^b (b-t) f^{(1)}(t) dt = \int_a^b f^{(1)}(t) (b-t) dt = \int udv = \left. uv \right| - \int vdu \\ &= \left. -f^{(2)}(t) \frac{(b-t)^2}{2} \right|_a^b -\int_a^b -\frac{(b-t)^2}{2}f^{(3)} (t)dt \\&= f^{(2)}(a) \frac{(b-a)^2}{2} + \int_a^b \frac{(b-t)^2}{2}f^{(3)}(t)dt \\ &= f^{(2)}(a) \frac{(b-a)^2}{2} + R_3 \end{split} \end{equation*}\]
n = m
\[ R_m = \int_a^b \frac{(b-t)^{m-1}}{(m-1)!} f^{(m)}(t)dt = \int_a^b f^{(m)}(t) \frac{(b-t)^{m-1}}{(m-1)!} dt = \int udv \] \[ u = f^{(m)}(t), \; dv = \frac{(b-t)^{m-1}}{(m-1)!} dt \] \[ du = f^{(m+1)}(t)dt, \; v = -\frac{(b-t)^m}{m!} \]
\[\begin{equation*} \begin{split} R_m &= \int_a^b \frac{(b-t)^{m-1}}{(m-1)!} f^{(1)}(t) dt = \int_a^b f^{(m)}(t) \ frac{(b-t)^{m-1}}{(m-1)!} dt \\ &= \int udv = \left. uv \right| - \int vdu \\ &= \left. -f^{(m)}(t) \frac{(b-t)^m}{m!} \right|_a^b -\int_a^b -\frac{(b-t)^m}{m!}f^{(m+1)}(t)dt \\ &= f^{(m)}(a) \frac{(b-a)^m}{m!} + \int_a^b \frac{(b-t)^m}{m!}f^{(m+1)}(t)dt \\ &= f^{(m)}(a) \frac{(b-a)^m}{m!} + R_{m+1} \end{split} \end{equation*}\]
\(R_n\) : Diverge or Converge?
if (b - a) < 1, then very rapidly \(\lim{n \to \infty} R_n = 0\)
Theorem
\[\lim_{n \to \infty} \frac{c^n}{n!} = 0\]
for any real number cProof. Let \(n_0\) be 2c < \(n_0\) < n, then c/\(n_0\) < 1/2
\[\begin{equation*} \begin{split} \frac{c^n}{n!} &= \frac{c \cdot c \cdot c \cdots c}{1 \cdot 2 \cdot 3 \cdots n} \\ &= \frac{c \cdot c \cdot c \cdots c \cdot c \cdots c}{1 \cdot 2 \cdot 3 \cdots n_0 \cdot (n_0 + 1) \cdots n} < \frac{c^{n_0}}{n_0 !} \frac{1}{2} \frac{1}{2} \cdots \frac{1}{2} \\ &= \frac{c^{n_0}}{n_0 !} \left( \frac{1}{2} \right)^{n - n_0} \end{split} \end{equation*}\]
\[ \lim_{n \to \infty} \frac{c^{n_0}}{n_0 !} \left( \frac{1}{2} \right)^{n - n_0} = 0 \]
\[\therefore \lim_{n \to \infty} \frac{c^n}{n!} = 0\] for any real number c
Exercise
Derive Taylor series for exp(x), log(1 + x), sin(x) + cos(x) with a=0, b=x.
Find yourself multivariate version of Taylor series.