$\hspace{.6em}$Went over Matlab lesson #1.

• Some special techniques for obtaining Power Series:
  $\mm$
  1. Basic operations: Multiplication and division.
  2. Substitution.
  3. Integration and Differentiation.
• Putting our methodology together:
  (1) Example: Find Maclaurin Series for $f(x) = \arctan x$.
  $\hspace{-1.2em}$ Solution.
  1. $\frac1{1+t^2} =\frac1{1-(-t^2)} = 1 + (-t^2) + (-t^2)^2 + (-t^2)^3 + \cdots $
  2. $ \arctan x = \arctan t \Bigg|_0^x = \int_0^x \frac{dt}{1+t^2} = t - \frac{t^3}3 + \frac{t^5}5-\frac{t^7}7 + \cdots \Bigg|_0^x $
  3. $ \arctan x = x - \frac{x^3}3 + \frac{x^5}5-\frac{x^7}7 + \cdots $
  
  (2) Example: Find Maclaurin Series for $f(x) = \frac{\sin \sqrt x}{\sqrt x}, \m x>0$.
  $\hspace{-1.2em}$ Solution.
  1. $ \frac1t \sin t = \frac1t \Bigg[ t -\frac{t^3}{3!}+ \frac{t^5}{5!} - \frac{t^7}{7!} + \cdots \Bigg] = 1 -\frac{t^2}{3!}+ \frac{t^4}{5!} - \frac{t^6}{7!} + \cdots$
  2. Substituting $t= \sqrt x$ into above, we find $\frac{\sin \sqrt x}{\sqrt x} = 1 -\frac{x}{3!}+ \frac{x^2}{5!} - \frac{x^3}{7!} + \cdots, \m 0< x <\infty.$

14. Error estimates:
    1. Error estimates for power serise: $\fcolorbox{white}{yellow}{$R_n(x) = f(x) -\Big[ f(a) + (x-a)f'(a) + \frac12 (x-a)^2 f''(a)+\cdots+ \frac1{n!} (x-a)^n f^{(n)}(a)\Big]$ }$
       $\mm\mm$
       $ R_n(x) = \frac{(x-a)^{n+1}}{(n+1)!} f^{(n+1)}(c)$, where $c$ is between $x$ and $a$.
    2. Alternating Series:
       $\fcolorbox{white}{yellow}{ $S= \sum_{n=1}^\infty a_n$ is alternating and $|a_{n+1}|< |a_n|$, $\lim_{n\to\infty} a_n = 0$.} $ Then $|S -(a_1+a_2+\cdots a_n)| \le |a_{n+1}|.$