Question

Find the Taylor series for

Answer 1

The Taylor Series expansion of f(x) = sin(x) about a = π i given by
$f(x)=\Sigma _{n=0}^{\infty} \,c_{n} (x-\pi)^{n}$
where the c's are contants.

That is 
f(x) = c₀ + c₁(x-π) +c (x-π)² + c₃ (x-π)³ + ...,
 ₂
The first few derivatives of f(x) are
f' = c₁
f'' = 2c₂ = 2! c₂
f''' = 3.2c₃ = 3! c₃
f⁽⁴⁾ = 4.3.2c₄ = 4! c₄
and so on.

The pattern indicates that
$c_{n}= \frac{f^{n}(\pi)}{n!}$

The derivatives of f(x) are
f' = cos(x)
f'' = -sin(x)
f''' = -cos(x)
f⁽⁴⁾ = sin(x(
and so on

The pattern indicates that
f⁽ⁿ⁾(x) = cos(x), n=1,5,9, ...,
         = -sin(x), n=2,6,10, ...,
         = -cos(x), n=3,7,11, ...,
         = sin(x), n=4,8,12, ...,

The radius of convergence is |x-π|<1 by the ratio test.