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}

tinfoQ 2020-12-17 Answered
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. \(f(x)=(1-x)^{-2}\)

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dessinemoie
Answered 2020-12-18 Author has 12716 answers

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
\(f(x)=\sum_{n=0}^{\infty}(n+1)x^n,R=1\)

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