Question

# Find the Maclaurin series for using the definition of a Maclaurin series. [Assume that has a power series expansion.Do not show that Rn(x) tends to 0.] Also find the associated radius of convergence. f(x)=(1-x)^{-2}

Series
Find the Maclaurin series for using the definition of a Maclaurin series. [Assume that has a power series expansion.Do not show that Rn(x) tends to 0.] Also find the associated radius of convergence. $$f(x)=(1-x)^{-2}$$

2020-12-18

Find a few derivatives, and calculate their values at a=0. Remember that there is a (-1) that comes from the derivative of (1-x)
$$f(x)=(1-x)^{-2}$$
$$f'(x)=-2(1-x)^{-3}(-1)$$
$$=2(1-x)^{-3}$$
$$f''(x)=(-3)2(1-x)^{-4}(-1)$$
$$=(3)(2)(1-x)^{-4}$$
$$f'''(x)=(-4)(3)2(1-x)^{-5}(-1)$$
$$=(4)(3)(2)(1-x)^{-5}$$
$$f^((4))(x)=(-5)(4)(3)(2)(1-x)^{-6}(-1)$$
$$=(5)(4)(3)(2)(1-x)^{-5}$$
$$f^n(x)=(n+1)!(1-x)^{-(n+2)}$$
Plug everything into Maclaurin general form:
$$f(x)=f(a)+\frac{f'(a)}{1!}x+\frac{f''(a)}{2!}x^2+\frac{f'''(a)}{3!}x^3+...$$
$$f(x)=1+2x+\frac{3!}{2!}x^2+\frac{4!}{3!}x^3+\frac{5!}{4!}x^4+...$$
$$=1+2x+3x^2+4x^3+5x^4+...$$
Find the pattern of the numbers to write in summation form
$$f(x)=\sum_{(n=0)}^\infty(n+1)x^n$$ Use the Ratio Test $$|\frac{a_{n+1}}{a_n}|=|\frac{((n+1)+1)x^{n+1}}{(n+1)x^n}|=|\frac{(n+2)x^{n+1}}{(n+1)x^n}|=\frac{(n+2)|x|}{n+1}$$
$$\lim_{n\rightarrow\infty}\frac{(n+2)|x|}{(n+1)}\cdot\frac{\frac{1}{n}}{\frac{1}{n}}=\lim_{n\rightarrow\infty}\frac{(1+\frac{2}{n})|x|}{1+\frac{1}{n}}=\frac{(1+0)|x|}{(1+0)}=|x|$$
This converges when
$$|x|<1$$
The radius of convergence is half the width of this interval
$$R=\frac{1-(-1)}{2}=1$$
Result
$$f(x)=\sum_{n=0}^{\infty}(n+1)x^n,R=1$$