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Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn(x) ? 0.]f(x) = 10/x , a= -2f(x) = \sum_{n=0}^{\infty } ______Find the associated radius of convergence R.R = ______

Respuesta :

LammettHash

Rewrite [tex]f[/tex] as

[tex]f(x)=\dfrac{10}x=-\dfrac5{1-\frac{x+2}2}[/tex]

and recall that for [tex]|x|<1[/tex], we have

[tex]\displaystyle\frac1{1-x}=\sum_{n=0}^\infty x^n[/tex]

so that for [tex]\left|\dfrac{x+2}2\right|<1[/tex], or [tex]|x+2|<2[/tex],

[tex]f(x)=-5\displaystyle\sum_{n=0}^\infty\left(\frac{x+2}2\right)^n[/tex]

Then the radius of convergence is 2.