Question

a) Find Taylor's series for the following about the point $x = a$ (or generated by $f$ at $x=a)$.

i) $f(x) = cosx, x = \frac{\pi}{4}$ ii) $f(x) = \frac{1}{1-x}, a = 0.$

iii) $f(x) = sin^2(x), a = 0.$ iv) $f(x) = 2^x, a = 1.$

v) $f(x) = x^2ln(x), a = 1$ vi) $f(x) = cos (3x), a = \frac{\pi}{3}.$

vii) $f(x) = ln(1+x), a = 0$ viii) $f(x) = ln(1-x^2), a = 0.$

ix) $f(x) = x^2ln(1+x^3), a = 0$

Answer 1

The question asks for Taylor series expansions of several functions. Each part requires a specific approach, often involving known series expansions or differentiation and evaluation at the given point.

Answer 2

The solution provides detailed steps for finding the Taylor series for each of the nine given functions. It utilizes the general Taylor series formula, known Maclaurin series, trigonometric identities, and algebraic manipulations to derive the series expansions.