How do you use the binomial series to expand  1/(2+x)^5?

Feb 7, 2016

Don't always use the Binomial theorem to expand because you can make mistakes.So,use the pascal's triangle.  See this pattern!!!! In the first term $a$ is raised to the maximum power of the equation and the power decreases as we go next to each term.This also happens for $b$,its power decreases when we come in the reverse direction.

So,

First solve ${\left(2 + x\right)}^{5}$

${\left(a + b\right)}^{5} = 1 {a}^{5} + 5 {a}^{4} b + 10 {a}^{3} {b}^{2} + 10 {a}^{2} {b}^{3} + 5 a {b}^{4} + 1 {b}^{5}$

Then,${\left(2 + x\right)}^{5} =$

$\rightarrow 1 {\left(2\right)}^{5} + 5 {\left(2\right)}^{4} \left(x\right) + 10 {\left(2\right)}^{3} \left({x}^{2}\right) + 10 {\left(2\right)}^{2} {\left(x\right)}^{3} + 5 \left(2\right) {\left(x\right)}^{4} + 1 {\left(x\right)}^{5}$

$\rightarrow 1 \left(32\right) + 5 \left(16\right) \left(x\right) + 10 \left(8\right) {\left(x\right)}^{2} + 10 \left(4\right) {\left(x\right)}^{3} + 5 \left(2\right) {\left(x\right)}^{4} + 1 {\left(x\right)}^{5}$

$\rightarrow 32 + 80 x + 80 {x}^{2} + 40 {x}^{3} + 10 {x}^{4} + {x}^{5}$

So,

$\implies \frac{1}{2 + x} ^ 5 = \frac{1}{32 + 80 x + 80 {x}^{2} + 40 {x}^{3} + 10 {x}^{4} + {x}^{5}}$