Question

Find taylor's series expansion for log(cosx) about the point x=pi/3.​

Answer 1

Required Series :

$-log(2)-\frac{1}{\sqrt{3}*1!}(x-\frac{\pi}{3})+\frac{4}{3*2!}(x-\frac{\pi}{3})^2+.....$

Steps:

1) Taylor series for any function f(x) = log(cos(x)) about any point a = pi/3 is given by

$f(a)+\frac{f'(a)}{1!}(x-\frac{\pi}{3})+\frac{f''(a)}{2!}(x-\frac{\pi}{3})^2+...$

Here,

$f(a) = log(cos ((\frac{\pi}{3}))=-log(2)\\ \\ f'(a)=\frac{1}{cos(x)}*(-sin(x))=-cot(\frac{\pi}{3})=-\frac{1}{\sqrt{3}} \\  \\f''(a)=cosec^2(\frac{\pi}{3})=\frac{4}{3}$

Then,

Our Taylor Series is given by

$f(x)=log(cos(x))=-log(2)-\frac{1}{\sqrt{3}*1!}(x-\frac{\pi}{3})+\frac{4}{3*2!}(x-\frac{\pi}{3})^2+.....$

Similarly,we can also derive series expansions of common functions like sine,cosine,exponential,etc.

Answer 2

Answer:

Step-by-step explanation: