The problem is to compute the sum of the series:

$S = \sum_{k=0}^{h} k \cdot 2^k$

To solve this, we use the following approach:

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Step 1: Derive the formula for the series

The summation involves $k \cdot 2^k$, and its exact summation formula can be derived using algebraic manipulation and calculus. The result for the sum is:

$S = (h - 1) \cdot 2^{h+1} + 2$

This formula is valid for $h \geq 1$. If $h = 0$, the sum is simply $0 \cdot 2^0 = 0$.

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Step 2: Compute for given $h$

Substitute the desired value of $h$ to calculate $S$. Let me know if you'd like to calculate for a specific value of $h$, or you can specify further constraints or changes to the series.

Would you like me to show the derivation of the formula, verify it with an example, or calculate for a specific $h$?

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Here are related questions to expand on this:

1. How is the formula for this summation derived?
2. Can we compute the same sum using programming or computational tools?
3. What happens if the limits change (e.g., $k = 1$ to $h$)?
4. How is this related to geometric or arithmetic series?
5. Are there alternative methods to derive this summation?

Tip: Understanding series summations often requires a combination of basic algebra, calculus, and pattern recognition.