Use the binomial series to expand the function as a power series. 9/(6 + x)³

Solution:

The power series can be rewritten as 9(x + 6)^-3.

 Since the power series has a negative power, the formula is given as :

\( (1+x)^{m} = 1 + mx + \frac{m(m -1)}{1.2}x^{2} + \frac{m(m - 1)(m - 2))}{1.2.3}x^{2}+ ,......... \)

And it holds whenever IxI < 1

f(x) can be written as:

f(x) = 9/[6(1+x/6)]³

= (9/6³)[1/(1 + x/6)³]

= (9/6³)(1 + x/6)^-3

m = -3 and x = x/6

Therefore the function f(x) can be expanded as follows:

f(x) = (9/6³)[ 1 + (-3)(x/6) +[ (-3)(-3-1)/2!](x/6)² + [(-3) (-3 - 1) (- 3 - 2) / 3!] (x / 6)³ + [(-3) (-3 - 1) (-3 - 2) (-3 - 3) / 4!] (x/6)⁴ + ……]

= (9/6³) [1 - x/2 + (1/6)x² - (1/18)x³ + (5/432)x⁴ + ……..

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Use the binomial series to expand the function as a power series. 9/(6 + x)³

Summary:

As a power series we get the following series as the expanded form of the function:

9/(6 + x)³ = (9/6³)[1 - x/2 + (1/6)x² - (1/18)x³ + (5/432)x⁴ + ……..]