Question

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

Answer 1

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 `(2+x)^5`

`(a+b)^5=1a^5+5a^4b+10a^3b^2+10a^2b^3+5ab^4+1b^5`

Then,`(2+x)^5=`

`rarr1(2)^5+5(2)^4(x)+10(2)^3(x^2)+10(2)^2(x)^3+5(2)(x)^4+1(x)^5`

`rarr1(32)+5(16)(x)+10(8)(x)^2+10(4)(x)^3+5(2)(x)^4+1(x)^5`

`rarr32+80x+80x^2+40x^3+10x^4+x^5`

So,

`=>1/(2+x)^5=1/(32+80x+80x^2+40x^3+10x^4+x^5)`