Find the coefficient of $x$ in the given polynomial

Question

We have polynomial $$(x-2^0)(x-2^1)(x-2^2)···(x-2^n)$$ I know that the coefficient of $x$ equals to: $$(-1)^{n-1}((2^0×2^1×···×2^{n-1})+(2^0×2^1×···×2^{n})+···+(2^1×2^2×···×2^{n}))$$

But it's hard to calculate this sum.

Any help is appreciated.

Answer 1

The coefficient of $x$ in $(x-a_1)\cdots(x-a_n)$ is $$ (-1)^n \sum_{k=1}^n \frac{a_1 \cdots a_n}{a_k} $$ If $a_k=2^{k-1}$, then $a_1 \cdots a_n = 2^0 2^1 \cdots 2^{n-1} = 2^{0+1+\cdots +(n-1)}=2^{n(n-1)/2}$.

Thus, the coefficient is $$ (-1)^n \sum_{k=1}^n \frac{2^{n(n-1)/2}}{2^{k-1}} = (-1)^n 2^{n(n-1)/2} \sum_{k=1}^n \frac{1}{2^{k-1}} = \cdots $$