Which of the following is 𝑍[𝑎𝑛]𝑧𝑛≥0Z[a n ]zn≥0?a)𝑧𝑧−𝑎z−az​ b)𝑧𝑧+𝑎−1z+a−1z​ c)𝑧𝑧+𝑎z+az​ d)𝑧−𝑎𝑧zz−a​

To determine which of the given expressions belong to the set $Z[a_n]z_{n \geq 0}$, we begin by analyzing the options. This set consists of rational functions that must be non-negative for $z \geq 0$.

Step 1: Analyze Each Expression

1. Option (a): $\frac{z}{z-a}$

   • For this function to be non-negative, the denominator cannot be zero. Since it goes to zero at $z = a$ and is positive when $z > a$, we need $z$ to stay away from $a$. Thus, this option is valid for $z > a$.

2. Option (b): $\frac{z}{z+a-1}$

   • The denominator becomes zero at $z = 1 - a$. If $a$ is greater than or equal to 1, this expression can also be valid. However, if $a < 1$ and $z \geq 0$, the expression can lead to negative values, so this option is conditionally valid.

3. Option (c): $\frac{z}{z+a}$

   • The denominator is never zero since $z + a > 0$ for $z \geq 0$ and $a \geq 0$. This expression is always non-negative for $a \geq 0$.

4. Option (d): $\frac{z-a}{z}$

   • This function can be negative when $z < a$. Thus, this expression does not maintain non-negativity for all $z \geq 0$.

Step 2: Summary of Findings

• Option (a) can only be non-negative for $z > a$.
• Option (b) is conditionally valid depending on $a$.
• Option (c) is always non-negative as long as $a \geq 0$.
• Option (d) can be negative for $z < a$.

Final Answer

The expression that fits within $Z[a_n]z_{n \geq 0}$ for all values of $z \geq 0$ and is guaranteed to be non-negative is (c) $\frac{z}{z+a}$.