- #1

olyviab

- 11

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## Homework Statement

Write the Taylor series of the function f(x) = ([tex]\pi[/tex] -x)^-2 around a = 0

## Homework Equations

([tex]\pi[/tex] - x)^-2 = f(a) + f'(a)(x-a) + [f''(a)(x-a)^2]/(2!) +......+ [f^n(a)(x-a)^n]/(n!)

## The Attempt at a Solution

This is what i have and i am not sure i am showing it correctly or compleatly.

f(x) = ([tex]\pi[/tex] -x)^-2

f'(x) = 2([tex]\pi[/tex] - x)^-3

f''(x) = 6([tex]\pi[/tex] - x)^-4

f'''(x) = 24([tex]\pi[/tex] - x)^-5

([tex]\pi[/tex] - x)^-2 = f(a) + [f'(a)(x-a)^2]/(2!) +......+ [f^n(a)(x-a)^n]/(n!)

= ([tex]\pi[/tex]-0)^-2 + [(2([tex]\pi[/tex]-0)^-3)(x-0)^2]/(2!) + [(6([tex]\pi[/tex]-0)^-4)(x-0)^3]/(3!) + ......

= [tex]\pi[/tex]^-2 + 2([tex]\pi[/tex]^-3)x +[6([tex]\pi[/tex]^-4)(x^2)]/(2!) + [24([tex]\pi[/tex]^-5)(x^3)]/(3!) + ........

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