- #1

- 1,435

- 187

Knopp said:1. If ##\prod_{n=1}^{\infty}(1+a_n)## is given, then this product, if we write

$$\prod_{\nu =1}^{n}(1+a_{\nu})=\mathfrak{p}_n \, ,$$

represents essentially the sequence ##(\mathfrak{p}_n )##. This sequence, on the other hand, is represented by the series

$$\mathfrak{p}_1 + ( \mathfrak{p}_2 - \mathfrak{p}_1) + ( \mathfrak{p}_3 - \mathfrak{p}_2) + \cdots = \mathfrak{p}_1+\sum_{n=2}^{\infty}(1+a_1)\cdots (1+a_{n-1})\cdot a_n$$

... [omitted text]

2. If conversely the series ##\sum_{n=1}^{\infty}a_n## is given, then it represents the sequence for which ##s_n=\sum_{\nu = 1}^{n}a_{\nu}##. This is also what is meant by the product

$$s_1\cdot\tfrac{s_2}{s_1}\cdot\tfrac{s_3}{s_2}\cdots = s_1\cdot \prod_{n=2}^{\infty}\tfrac{s_n}{s_{n-1}} = a_1\cdot \prod_{n=2}^{\infty}\left( 1+\tfrac{a_n}{a_1+a_2+\cdots +a_{n-1}}\right) \, ,$$

--provided it has meaning at all. ...

Maybe that'll break the ice... please post your favorite infinite product (or application thereof).