Expanding a product formally.

Question

Let $a_1,...,a_n$ be real numbers.

I don't know how to formally expand the following product

$$ \prod_{k=1}^n(1+a_k) $$

I'm guessing something like (edited) $$1+\huge\sum_{k=1}^n \; \huge\sum_{1\leq j_1<...<j_k\leq n}a_{j_1}...a_{j_k}$$

But I'm not sure. Can someone check ?

Answer 1

You have to add $1$ to the sum. You obtain it selecting $1$ in every term of the product.

Perhaps you prefer this expression. Let be $S=\{1,\ldots,n\}$. Then, the expansion gives: $$\sum_{T\subset S}\prod_{k\in T} a_k$$ The term you have forgotten is when $T=\emptyset$.