Products of Functions that Don't Depend on Index

Question

Is there a general property of products that allows you to simplify

$$ \prod_{i = 1}^n f(x) \, g(i) $$

where $f(x)$ does not depend on $i$? Would it just be

$$ f^{n}(x) \prod_{i = 1}^n g(i) $$

since you can just move $f$ outside of the product?

Answer 1

Yes, that is exactly what it would be.

Answer 2

Careful with the notation. $f^n(x)$ is a bit ambiguous. It could refer to the $n^\text{th}$ derivative of $f$, or the function $f$ applied to $x$, $n$ times. In this case, I think you meant to write $\left[f(x)\right]^n$.

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To answer your question, if $f(x)$ is independent of the index $i$, then $$\displaystyle\prod\limits_{i=1}^n f(x)g(i) = \left[f(x)\right]^n \displaystyle\prod\limits_{i=1}^n g(i)$$

I hope that helps.

Answer 3

The answer is virtually self-evident if you work on it without the "$\prod$" notation: $$ \begin{align} \prod_{i=1}^3 (x^5 i^2) & = (x^5 1^2)\cdot(x^5 2^2)\cdot(x^5 3^2) \\[15pt] & = (x^5)^3\Big(1^2\cdot2^2\cdot3^2\Big) \\[15pt] & = (x^5)^3\prod_{i=1}^3 i^2. \end{align} $$