Thread: Binomial Expansions

Q1)Find, in its simplest form, the coefficient of in the expansion, in ascending powers of , of

Q2) Expand as a series of ascending powers of up to and including the term in , Simplify the coefficients.

I got the above part right and derived at the answer of

However, its the following part that i dont get

In The expansion of , where is a constant, the coefficient of is zero. By writing as , or otherwise, find the value of k.

Any help is appreciated ^^

Originally Posted by teddybear67

Q1)Find, in its simplest form, the coefficient of in the expansion, in ascending powers of , of

Q2) Expand as a series of ascending powers of up to and including the term in , Simplify the coefficients.

I got the above part right and derived at the answer of

However, its the following part that i dont get

In The expansion of , where is a constant, the coefficient of is zero. By writing as , or otherwise, find the value of k.

Any help is appreciated ^^

hi

begin with some substitution as hinted by the question

1+14(x+kx^2)+91(x+kx^2)^2+364(x+kx^2)^3+91(x^2+2kx ^3+k^2x^4)+364(x^3+3kx^4+3x(kx^2)^2+(kx^2)^3)

Given that the coefficient of is 0 , so we need to know the coefficient of x^3 only . To expand the whole thing would be tedious , so we expand those parts where is .



coefficient of x^3 :

Thanks!
But do ya know how to do the top one too? :/

Use the "generalized binomial theorem"

Where, for r not a positive integer,

With r not a positive integer, r- i+ 1 is never equal to 0 so this is an infinite series.