Find-the-values-of-k-and-n-for-which-x-3-and-higher-powers-of-x-are-negligeble-given-that-1-kx-n-1-2x-6x-2-

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

Find the values of k and n for which x^{3} and higher powers of x are negligeble

given that {(1 + kx)}^{n} = 1 + 2 x + {6 x}^{2} .

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

{(1 + k x)}^{n} = 1 + n k x + \frac{n (n - 1) k^{2} x^{2}}{2} + o (x^{3})

n k = 2

\frac{n (n - 1) k^{2}}{2} = 6

\Rightarrow n = - \frac{1}{2}

\Rightarrow k = - 4

\frac{1}{\sqrt{1 - 4 x}} = 1 + 2 x + 6 x^{2} + o (x^{3})

b u t x^{3} and higher powers of x are negligeble

o n l y i f x \to 0 .

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

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