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Question Number 17957 by Arnab Maiti last updated on 13/Jul/17

prove that (1+(a/x))^n =(1+((an)/x)) , x≫a

$$\mathrm{prove}\:\mathrm{that}\:\left(\mathrm{1}+\frac{\mathrm{a}}{\mathrm{x}}\right)^{\mathrm{n}} =\left(\mathrm{1}+\frac{\mathrm{an}}{\mathrm{x}}\right)\:,\:\:\mathrm{x}\gg\mathrm{a} \\ $$

Answered by 42 last updated on 13/Jul/17

Expand using binomial theorem: 1 + n((a/x)) + ((n(n − 1))/2)((a/x))^2 + ... Since a ≪ x so ((a/x))^2 , ((a/x))^3 , ... are so small and can be neglected. ∴(1 + (a/x))^n = 1 + ((an)/x), provided a ≪ x

$$\mathrm{Expand}\:\mathrm{using}\:\mathrm{binomial}\:\mathrm{theorem}: \\ $$$$\mathrm{1}\:+\:{n}\left(\frac{{a}}{{x}}\right)\:+\:\frac{{n}\left({n}\:−\:\mathrm{1}\right)}{\mathrm{2}}\left(\frac{{a}}{{x}}\right)^{\mathrm{2}} \:+\:... \\ $$$$\mathrm{Since}\:{a}\:\ll\:{x}\:\mathrm{so}\:\left(\frac{{a}}{{x}}\right)^{\mathrm{2}} ,\:\left(\frac{{a}}{{x}}\right)^{\mathrm{3}} ,\:...\:\mathrm{are}\:\mathrm{so} \\ $$$$\mathrm{small}\:\mathrm{and}\:\mathrm{can}\:\mathrm{be}\:\mathrm{neglected}. \\ $$$$\therefore\left(\mathrm{1}\:+\:\frac{{a}}{{x}}\right)^{{n}} \:=\:\mathrm{1}\:+\:\frac{{an}}{{x}},\:\mathrm{provided}\:{a}\:\ll\:{x} \\ $$

Commented by Arnab Maiti last updated on 19/Jul/17

Please solve the question without using binomial theorem.

$$\mathrm{Please}\:\mathrm{solve}\:\mathrm{the}\:\mathrm{question}\: \\ $$$$\mathrm{without}\:\mathrm{using}\:\mathrm{binomial}\:\mathrm{theorem}. \\ $$

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