Find the Taylor polynomial T3(x) for the function f(x)=1/x centered at the number a=4?

I feel like I’m doing it right but I keep getting the wrong answer, can someone please work this out so I can see if it’s the website or if it’s me? (the site has glitched before to make right answers wrong).

Answer 1

f(x) = 1/x = x^(-1)
f'(x) = – x^(-2)
f”(x) = 2x^(-3)
f”'(x) = – 6 x^(-4)

f(4)=1/4
f'(4)=-1/16
f”(4)= 1/32
f”'(4) = – 1/1536

f(x) = f(4) + f'(4)(x – 4) + f”(4)(x – 4)^2/2 + f”'(4)(x – 4)^3/(3*2)

f(x) = 1/4 – (x – 4)/16 + (x – 4)^2/64 – (x – 4)^3/256

Answer 2

the third term would be :

f ‘ ‘ ‘ (4) / 4! * ( x – 4 )^3 / 3 !

f ‘ ‘ ‘ (4) = -3 / 128

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