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Sagot :
The taylor series for the f(x)=8/x centered at the given value of a=-4 is -2+2(x+4)/1!-24/16 [tex](x+4)^{2}[/tex]/2!+...........
Given a function f(x)=9/x,a=-4.
We are required to find the taylor series for the function f(x)=8/x centered at the given value of a and a=-4.
The taylor series of a function f(x)=[tex]f(a)+f^{1}(a)(x-a)/1!+ f^{11}(a)(x-a)^{2} /2! +f^{111}(a)(x-a)a^{3}/3!+..........[/tex]
Where the terms in f prime [tex]f^{1}[/tex](a) represent the derivatives of x valued at a.
For the given function.f(x)=8/x and a=-4.
So,f(a)=f(-4)=8/(-4)=-2.
[tex]f^{1}[/tex](a)=[tex]f^{1}[/tex](-4)=-8/([tex]-4)^{2}[/tex]
=-8/16
=-1/2
The series of f(x) is as under:
f(x)=f(-4)+[tex]f^{1}(-4)(x+4)/1!+ f^{11}(-4)(x+4)^{2}/2!.............[/tex]
[tex]=8/(-4)-8/(-4)^{2} (-4)(x+4)/1!+ 24/(-4)^{3} (-4)(x+4)^{2}/2!.............[/tex]
=-2+2(x+4)/1!-24/16 [tex](x+4)^{2}[/tex]/2!+...........
Hence the taylor series for the f(x)=8/x centered at the given value of a=-4 is -2+2(x+4)/1!-24/16 [tex](x+4)^{2}[/tex]/2!+...........
Learn more about taylor series at https://brainly.com/question/23334489
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