5.12 テイラー級数の計算

5.25 (テイラー級数の計算例)  

$\displaystyle \frac{1}{1-x}$ $\displaystyle =1+x+x^2+x^3+\cdots$    
$\displaystyle \frac{1}{1-x-x^2}$ $\displaystyle = 1+\left(x+x^2\right)+ \left(x+x^2\right)^2+ \left(x+x^2\right)^3+\cdots$    
  $\displaystyle =1+(x+x^2)+(x^2+2x^3+x^4)+(x^3+3x^4+3x^5+x^6)+\cdots$    
  $\displaystyle =1+x+2x^2+3x^3+4x^4+\cdots$    

5.26 (テイラー級数の計算例)  

$\displaystyle (1+x)^{\frac{1}{2}}$ $\displaystyle =1+\frac{1}{2}x+ \frac{\frac{1}{2}\left(-\frac{1}{2}\right)}{2}x^...
...c{\frac{1}{2}\left(-\frac{1}{2}\right)\left(-\frac{1}{3}\right)}{3!}x^3+ \cdots$    
  $\displaystyle = 1+\frac{1}{2}x-\frac{1}{8}x^2+\frac{1}{16}x^3+\cdots$    
$\displaystyle \sqrt{1-x}$ $\displaystyle = 1-\frac{1}{2}x-\frac{1}{8}x^2-\frac{1}{16}x^3+\cdots$    
$\displaystyle \sqrt{1-x-x^2}$ $\displaystyle = 1-\frac{1}{2}(x+x^2)-\frac{1}{8}(x+x^2)^2-\frac{1}{16}(x+x^2)^3+\cdots$    
  $\displaystyle = 1-\frac{1}{2}x-\left(\frac{1}{2}+\frac{1}{8}\right)x^2- \left(\frac{2}{8}+\frac{1}{16}\right)x^3+\cdots$    
  $\displaystyle = 1-\frac{1}{2}x-\frac{5}{8}x^2-\frac{5}{16}x^3+\cdots$    

5.27 (テイラー級数の計算例)  

$\displaystyle e^{x}$ $\displaystyle = 1+x+\frac{x^2}{2}+\frac{x^3}{3!}+\cdots= \sum_{n=0}^{\infty}\frac{x^n}{n!}$    
$\displaystyle e^{-x}$ $\displaystyle = 1-x+\frac{x^2}{2}-\frac{x^3}{3!}+\cdots= \sum_{n=0}^{\infty}\frac{(-1)^n x^n}{n!}$    
$\displaystyle \sinh x$ $\displaystyle =\frac{1}{2}(e^{x}-e^{-x})$    
  $\displaystyle =\frac{1}{2}\left( \sum_{n=0}^{\infty}\frac{x^n}{n!} - \sum_{n=0}...
...{(-1)^n x^n}{n!} \right) = \frac{1}{2}\sum_{n=0}^{\infty}\frac{1-(-1)^n}{n!}x^n$    
  $\displaystyle \quad(n=2k,n=2k+1;k=0,1,2,\cdots)$    
  $\displaystyle = \frac{1}{2}\sum_{k=0}^{\infty}\frac{1-(-1)^{2k}}{(2k)!}x^{2k}+ \frac{1}{2}\sum_{k=0}^{\infty}\frac{1-(-1)^{2k+1}}{(2k+1)!}x^{2k+1}$    
  $\displaystyle = \sum_{k=0}^{\infty}\frac{1}{(2k+1)!}x^{2k+1}$    
  $\displaystyle \quad(k\to n-1;n=1,2,3,\cdots)$    
  $\displaystyle = \sum_{n=1}^{\infty}\frac{1}{(2n-1)!}x^{2n-1}$    

5.28 (テイラー級数の計算)   $ \cosh x$ のテイラー級数を求めよ.

Kondo Koichi
平成19年1月23日