The expression in the image is a double summation, which can be interpreted as follows:

$\sum_{n=1}^3 \left( \sum_{n+1 < z \leq 4} \left( y_{n z} + \frac{n^3}{z} \right) \right)$

Step-by-Step Breakdown

1. Outer Summation:

   • The outer summation index is $n$, which goes from $n = 1$ to $n = 3$.
   • This controls the inner summation for each integer value of $n$ within that range.

2. Inner Summation:

   • The inner summation index is $z$.
   • For each $n$, $z$ ranges over integers that satisfy $n+1 < z \leq 4$.
   • This means the range for $z$ will depend on the value of $n$:
     • When $n = 1$: $z$ ranges from $z = 3$ to $z = 4$.
     • When $n = 2$: $z$ can only be $4$.
     • When $n = 3$: there is no valid $z$ satisfying $n+1 < z \leq 4$ (so this term is effectively zero).

3. Inner Expression:

   • Inside the inner summation, the expression being summed is $y_{n z} + \frac{n^3}{z}$.
   • $y_{n z}$ likely represents some variable or matrix indexed by $n$ and $z$, and $\frac{n^3}{z}$ is a fractional term based on $n$ and $z$.

Explicit Calculation

Given the ranges identified above, let’s compute the values step by step:

• For $n = 1$:

  • Possible values for $z$ are $3$ and $4$.
  • So, we calculate: $\sum_{z=3}^{4} \left( y_{1 z} + \frac{1^3}{z} \right) = \left( y_{1 3} + \frac{1}{3} \right) + \left( y_{1 4} + \frac{1}{4} \right)$

• For $n = 2$:

  • Possible value for $z$ is $4$ only.
  • So, we calculate: $\sum_{z=4}^{4} \left( y_{2 z} + \frac{2^3}{z} \right) = y_{2 4} + \frac{8}{4} = y_{2 4} + 2$

• For $n = 3$:

  • There are no values of $z$ that satisfy $3 + 1 < z \leq 4$, so this term contributes $0$.

Final Expression

Combining these results, the entire summation evaluates to:

$\left( y_{1 3} + \frac{1}{3} \right) + \left( y_{1 4} + \frac{1}{4} \right) + \left( y_{2 4} + 2 \right)$

Would you like a deeper explanation on any of these steps or help with a specific part of the summation?

Further Questions

1. How can we simplify expressions involving double summations like this one?
2. What does it mean if there are no valid terms in a summation?
3. How could you evaluate this expression if you had specific values for $y_{n z}$?
4. How do boundary conditions (like $z \leq 4$) affect the summation limits?
5. How would the result change if the upper limit of $n$ or $z$ was different?

Tip

When dealing with summations with variable limits, evaluate the ranges of each index carefully to ensure you capture all valid terms.