Among all possible representations of the positive integer $ n$ as $ n\equal{}\sum_{i\equal{}1}^k a_i$ with positive integers $ k, a_1 < a_2 < ...<a_k$, when will the product $ \prod_{i\equal{}1}^k a_i$ be maximum?
1. **Understanding the Problem:**
   We need to find the representation of a positive integer \( n \) as \( n = \sum_{i=1}^k a_i \) with positive integers \( k, a_1 < a_2 < \ldots < a_k \) such that the product \( \prod_{i=1}^k a_i \) is maximized.

2. **Analyzing the Product:**
   Consider two numbers \( a \) and \( a + k \) (where \( k > 1 \)). The product of these two numbers is:
   \[
   a(a + k) = a^2 + ak
   \]

3. **Reducing the Difference:**
   If we reduce the difference between these numbers while maintaining the sum, they become \( a + 1 \) and \( a + k - 1 \). The product of these new numbers is:
   \[
   (a + 1)(a + k - 1) = a^2 + ak + k - 1
   \]

4. **Comparing the Products:**
   We compare the two products:
   \[
   a^2 + ak \quad \text{and} \quad a^2 + ak + k - 1
   \]
   Clearly,
   \[
   a^2 + ak + k - 1 > a^2 + ak
   \]
   This shows that reducing the difference between the numbers increases the product.

5. **Minimizing the Difference:**
   To maximize the product \( \prod_{i=1}^k a_i \), we need to minimize the difference between the largest and smallest \( a_i \). This implies that the numbers \( a_i \) should be as close to each other as possible.

6. **Optimal Representation:**
   The optimal representation of \( n \) is when the numbers \( a_i \) are consecutive integers. This is because consecutive integers have the smallest possible difference between the largest and smallest numbers, thus maximizing the product.

Conclusion:
The product \( \prod_{i=1}^k a_i \) will be maximum when the numbers \( a_i \) are consecutive integers.

The final answer is the product will be maximum when the numbers \( \boxed{ a_i } \) are consecutive integers.