Question

Obtain the Maclaurin series for f(x)=(1/(2-x))

Answer 1

The maclaurin series is given by:

f(x)= f(0) + f'(0)x + $\frac {x^2 f''(0)}{2!}$ + $\frac {x^3 f'''(0)}{3!}$ + ..... +$\frac {x^n f^n(0)}{n!}+.....$

f = (2-x)^-1

f' = (2-x)^-2

f''' = -2(2-x)^-3

f'''' = 6(2-x)^-4

........

f(0) = 1/2

f'(0) = -2⁴

f''(0) = 6.2^-4

......

maclaurin series = $\frac {1}{2}$ + $\frac {x}{4} - \frac{x^2}{2^3} + \frac{x^3}{2^4} - \frac{x^4}{2^5}+ .....$