# 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}

= (7/4^{3})[1/(1 + x/4)^{3}]

= (7/4^{3})(1 + x/4)^{-3}

m = -3 and x = x/4

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

f(x) = (7/4^{3})[ 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} + ……]

= (7/4^{3})[1-3x/4 + (3/8)x^{2} - (15/32)x^{3} + (15/256)x^{4} + ……..]

## 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 } = = (7/4^{3})[1-3x/4 + (3/8)x^{2} - (15/32)x^{3} + (15/256)x^{4} + ……..]