Rewriting expressions using ∑?

Question

How would you write this expression using $\sum$ notation?

$(x + 1)(x − 3)(x + 5)(x − 7)· · ·(x + 101)$

Thanks!

Answer 1

$$(x + 1)(x − 3)(x + 5)(x − 7)· · ·(x + 101)=\prod_{n=0}^{50}(x-(-1)^{n+1}(2n+1))$$ I can use a trick $y=e^{\ln y}$: $$(x + 1)(x − 3)(x + 5)(x − 7)· · ·(x + 101)=e^{\ln \prod_{n=0}^{50}(x-(-1)^{n+1}(2n+1))}\\=e^{\sum_{n=0}^{50}\ln(x-(-1)^{n+1}(2n+1))}$$