i-a-n-i-for-i-lt-1-a-R-

Question Number 3198 by Filup last updated on 07/Dec/15

\underset{i = a}{\prod^{n}} i = ? for ∣ i ∣< 1, a \in R

Commented by Filup last updated on 07/Dec/15

= \frac{1}{a} \times \frac{1}{a + 1} \times \dots \times \frac{1}{n}

= \frac{1}{(\frac{n!}{(a - 1)!})}

= \frac{(a - 1)!}{n!}

correct ?

Commented by prakash jain last updated on 07/Dec/15

I {don}^{'} t think factorial function is

defined for real numbers .

Commented by Filup last updated on 07/Dec/15

W h a t about in Γ notation ?

Commented by prakash jain last updated on 07/Dec/15

Let us say n = i + k k \in Z^{+}

\underset{i = 1}{\prod^{n}} i = a (a + 1) (a + 2) \dots (a + k)

Commented by prakash jain last updated on 07/Dec/15

If n = i + k, k \in Z^{+}

then Γ (n) a n d Γ (i) be derived with

Γ (x + 1) = x Γ (x)