# Expanding a product formally.

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 ?

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you are missing a "one" –  rlartiga Apr 15 '14 at 13:04
@rlartiga right. Is the rest valid ? –  LeGrandDODOM Apr 15 '14 at 13:05
It should be $1 \leqslant j_1 < \dotsc < j_k \leqslant n$. Some products include the term $a_n$. –  Daniel Fischer Apr 15 '14 at 13:06

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 $T=\emptyset$.

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Did you mean $S={1,...,n}$ ? –  LeGrandDODOM Apr 15 '14 at 13:08
Yes, thank you. –  ajotatxe Apr 16 '14 at 12:31