Use the binomial series to expand the function as a power series. 7/(4 + x)3
Solution:
The power series can be rewritten as 7 /(4 + x)-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) = 7/[4(1+x/4)]3
= (7/43)[1/(1 + x/4)3]
= (7/43)(1 + x/4)-3
m = -3 and x = x/4
Therefore the function f(x) can be expanded as follows:
f(x) = (7/43)[ 1 + (-3)(x/4) +[ (-3)(-3-1)/2!](x/4)2 + [(-3)(-3-1)(-3-2)/3!](x/4)3 +[(-3)(-3-1)(-3-2)(-3-3)/4!](x/4)4 + ……]
= (7/43)[1-3x/4 + (3/8)x2 - (15/32)x3 + (15/256)x4 + ……..]
Use the binomial series to expand the function as a power series. 7/(4 + x)3
Summary:
As a power series we get the following series as the expanded form of the function:
7/(4 + x)3 = = (7/43)[1-3x/4 + (3/8)x2 - (15/32)x3 + (15/256)x4 + ……..]
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