Obtain-the-first-four-term-of-the-expansion-4-x-1-3-when-1-x-lt-1-ii-x-gt-1-

Question Number 88039 by peter frank last updated on 08/Apr/20

O b t a i n t h e f i r s t f o u r

t e r m o f t h e e x p a n s i o n

{(4 - x)}^{\frac{1}{3}} w h e n

(1) ∣ x ∣< 1

(i i) ∣ x ∣> 1

Answered by Rio Michael last updated on 08/Apr/20

(i) for ∣ x ∣ < 1,

{(4 - x)}^{\frac{1}{3}} = 4^{\frac{1}{3}} {(1 - \frac{x}{4})}^{\frac{1}{3}}

valid for ∣ \frac{x}{4} ∣ < 1 or ∣ x ∣ < 4 therefore above condition is valid .

4^{\frac{1}{3}} (1 + (- \frac{x}{4}) (\frac{1}{3}) + \frac{\frac{1}{3} (- \frac{2}{3})}{2!} {(- \frac{x}{4})}^{2} + \frac{\frac{1}{3} (- \frac{2}{3}) (- \frac{5}{3})}{6} {(- \frac{x}{4})}^{3} + \dots) =

4^{\frac{1}{3}} (1 - \frac{x}{12} - \frac{x^{2}}{144} - \frac{10 x^{3}}{10368} + \dots)

for (i i) the expansion is same as both conditions are valid

Commented by peter frank last updated on 15/Apr/20

t h a n k y o u