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Question Number 78332 by Lontum Hans last updated on 16/Jan/20

$$\mathrm{Find}\mathrm{the}\mathrm{values}\mathrm{of}\mathrm{k}\mathrm{and}\mathrm{n}\mathrm{for}\mathrm{which}{\mathrm{x}}^3\mathrm{and}\mathrm{higher}\mathrm{powers}\mathrm{of}\mathrm{x}\mathrm{are}\mathrm{negligeble}$$$$\mathrm{given}\mathrm{that}{(1+\mathrm{kx})}^{\mathrm{n}}=1+2\mathrm{x}+{6\mathrm{x}}^2.$$

Commented by mr W last updated on 16/Jan/20

$${(1+kx)}^n=1+nkx+\frac{n(n-1)k^2{x}^2}{2}+o\left(x^3\right)$$$$nk=2$$$$\frac{n(n-1)k^2}{2}=6$$$$\Rightarrow n=-\frac{1}{2}$$$$\Rightarrow k=-4$$$$\frac{1}{\sqrt{1-4x}}=1+2x+6x^2+o\left(x^3\right)$$$$$$$$but{\mathrm{x}}^3\mathrm{and}\mathrm{higher}\mathrm{powers}\mathrm{of}\mathrm{x}\mathrm{are}\mathrm{negligeble}$$$$onlyifx\to 0.$$

Commented by mr W last updated on 16/Jan/20

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