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1.4.1 Sequences and Series - The Binomial Theorem

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Question 88
a.

Use the binomial expansion to expand

(9−4x)−12∣x∣<94 (9 - 4x)^{-\frac{1}{2}} \quad |x| < \frac{9}{4} (9−4x)−21​∣x∣<49​

in ascending powers of xxx, up to and including the term in x2x^2x2, giving each coefficient as a fully simplified fraction.

[4]
b.
f(x)=6+kx9−4xwhere k is a constant and ∣x∣<94 f(x) = \frac{6 + kx}{\sqrt{9 - 4x}} \quad \text{where } k \text{ is a constant and } |x| < \frac{9}{4} f(x)=9−4x​6+kx​where k is a constant and ∣x∣<49​

Given that the series expansion of f(x)f(x)f(x), in ascending powers of xxx, is

2+49x+px2+…where p is a constant 2 + \frac{4}{9}x + px^2 + \dots \quad \text{where } p \text{ is a constant} 2+94​x+px2+…where p is a constant

find the value of kkk,

[2]
c.

find the value of ppp.

[2]

1.4.1 Sequences and Series - The Binomial Theorem Questions

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