Question

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

Answer 1

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

Answer 2

The given expression is a product:

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

This notation means we multiply terms of the form $(1+x^n)^n$ for each integer value of $n$ from $1$ to $k$.

Let's write out the terms of the product:

For $n=1$, the term is $(1+x^1)^1 = (1+x)$.

For $n=2$, the term is $(1+x^2)^2$.

For $n=3$, the term is $(1+x^3)^3$.

...

For $n=k$, the term is $(1+x^k)^k$.

So, the product can be written as:

$(1+x)^1 \cdot (1+x^2)^2 \cdot (1+x^3)^3 \cdot \ldots \cdot (1+x^k)^k$

This expression represents a polynomial in $x$ if $k$ is a positive integer. For example:

• If $k=1$, the product is $(1+x)$.
• If $k=2$, the product is $(1+x)(1+x^2)^2 = (1+x)(1+2x^2+x^4) = 1+x+2x^2+2x^3+x^4+x^5$.

This type of product does not simplify into a common closed-form expression using standard mathematical identities or formulas. The terms in the product, $(1+x^n)^n$, are distinct for different values of $n$ and do not exhibit a pattern that allows for cancellation (like in telescoping products) or combination into a simpler form (like in geometric series products).

Therefore, the given expression is already in its most compact and general form. There is no simpler algebraic closed-form solution for this product for arbitrary $k$ and $x$.

The expression itself is the 'solution' in the sense that it's the most simplified and general representation of the product.