Q: if n is a positive integer, find the coefficient of x^r in the expansion of (1+x)(1-x)^n as a series of ascending powers of x.

Solution: (1+x)(1-x)^n = (1-x)^n + x(1-x)^n

(1) = SUM(r=0 ->n) n_C_r(-x)^r + xSUM(r=0 ->n) n_C_r(-x)^r
(2) = SUM(r=0 ->n) n_C_r(-x)^r + SUM(r=0 ->n) n_C_r(-x)^(r+1)
(3) = (1 - n_C_1x + n_C_rx^2+...+ n_C_(r-1)(-1)^(r-1)(x)^(r-1)
+ n_C_r(-1)^(r)(x)^(r) + ... + (-1)^(n)(x)^(n)) + (x -
n_C_1x^2 + ... + n_C_(r-1)(-1)^(r-1)(x)^(r) +
n_C_r(-1)^(r)(x)^(r+1) + (-1)^(n)(x)^(n+1))
(4) = SUM(r=0 ->n) (n_C_r(-1)^r + n_C_(r-1)(-1)^(r-1))x^r

Then they say the term of x^r is n_C_r(-1)^r + n_C_(r-1)(-1)^(r-1). Now, it is in the last step of the equations that I have a problem, namely in equation (3) the highest degree of x is n+1, while in equation (4) the highest power of x is n. What happened to the term (-1)^(n)(x)^(n+1) ???

Thanks, Nico

That's correct assuming n_C_(-1) = 0.

>What happened to the term
>(-1)^(n)(x)^(n+1) ???
>

They've lost it. Indeed, the summation in (4) is only up to xⁿ.