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

Obtain the first four  term of the expansion  (4−x)^(1/3) when  (1)∣x∣<1  (ii)∣x∣>1

$${Obtain}\:{the}\:{first}\:{four} \\ $$ $${term}\:{of}\:{the}\:{expansion} \\ $$ $$\left(\mathrm{4}−{x}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} {when} \\ $$ $$\left(\mathrm{1}\right)\mid{x}\mid<\mathrm{1} \\ $$ $$\left({ii}\right)\mid{x}\mid>\mathrm{1} \\ $$

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

 (i) for ∣x∣ < 1,    (4−x)^(1/3)  = 4^(1/3) (1−(x/4))^(1/3)    valid for ∣(x/4)∣ < 1 or   ∣x∣ < 4 therefore above condition is valid.      4^(1/3) ( 1 + (−(x/4))((1/3)) + (((1/3)(−(2/3)))/(2!))(−(x/4))^2  + (((1/3)(−(2/3))(−(5/3)))/6)(−(x/4))^3 +...) =      4^(1/3) (1−(x/(12)) −(x^2 /(144))−((10x^3 )/(10368)) + ...)   for (ii) the expansion is same as both conditions are valid

$$\:\left({i}\right)\:\mathrm{for}\:\mid{x}\mid\:<\:\mathrm{1}, \\ $$ $$\:\:\left(\mathrm{4}−{x}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} \:=\:\mathrm{4}^{\frac{\mathrm{1}}{\mathrm{3}}} \left(\mathrm{1}−\frac{{x}}{\mathrm{4}}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} \: \\ $$ $$\mathrm{valid}\:\mathrm{for}\:\mid\frac{{x}}{\mathrm{4}}\mid\:<\:\mathrm{1}\:\mathrm{or}\:\:\:\mid{x}\mid\:<\:\mathrm{4}\:\mathrm{therefore}\:\mathrm{above}\:\mathrm{condition}\:\mathrm{is}\:\mathrm{valid}. \\ $$ $$\:\:\:\:\mathrm{4}^{\frac{\mathrm{1}}{\mathrm{3}}} \left(\:\mathrm{1}\:+\:\left(−\frac{{x}}{\mathrm{4}}\right)\left(\frac{\mathrm{1}}{\mathrm{3}}\right)\:+\:\frac{\frac{\mathrm{1}}{\mathrm{3}}\left(−\frac{\mathrm{2}}{\mathrm{3}}\right)}{\mathrm{2}!}\left(−\frac{{x}}{\mathrm{4}}\right)^{\mathrm{2}} \:+\:\frac{\frac{\mathrm{1}}{\mathrm{3}}\left(−\frac{\mathrm{2}}{\mathrm{3}}\right)\left(−\frac{\mathrm{5}}{\mathrm{3}}\right)}{\mathrm{6}}\left(−\frac{{x}}{\mathrm{4}}\right)^{\mathrm{3}} +...\right)\:=\: \\ $$ $$\:\:\:\mathrm{4}^{\frac{\mathrm{1}}{\mathrm{3}}} \left(\mathrm{1}−\frac{{x}}{\mathrm{12}}\:−\frac{{x}^{\mathrm{2}} }{\mathrm{144}}−\frac{\mathrm{10}{x}^{\mathrm{3}} }{\mathrm{10368}}\:+\:...\right)\: \\ $$ $$\mathrm{for}\:\left({ii}\right)\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{is}\:\mathrm{same}\:\mathrm{as}\:\mathrm{both}\:\mathrm{conditions}\:\mathrm{are}\:\mathrm{valid} \\ $$

Commented bypeter frank last updated on 15/Apr/20

thank you

$${thank}\:{you} \\ $$

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