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The question asks to find the sum of the infinite series given by:

\[
\sum_{n \geq 0} \frac{n^2}{2^n}
\]

This is an infinite series where each term is of the form \( \frac{n^2}{2^n} \). Let's solve it step by step.

### Step 1: Recognize the series structure
The series is a power series with a general term \( \frac{n^2}{2^n} \). Such series can often be computed using known formulas or through differentiation of simpler geometric series.

### Step 2: Utilize known series result
One way to handle series involving \( n^2 \) is by using the fact that:

\[
\sum_{n=0}^{\infty} \frac{n^2}{x^n} = \frac{x(x+1)}{(x-1)^3}
\]

By substituting \( x = 2 \) (since our series involves powers of 2), we get:

\[
\sum_{n=0}^{\infty} \frac{n^2}{2^n} = \frac{2(2+1)}{(2-1)^3} = \frac{6}{1^3} = 6
\]

### Final Answer:
The sum of the series is **6**, which corresponds to option **B**.

Would you like more details on the steps? Here are five related questions you might find helpful:

1. How can we derive the sum formula for series involving \( n^2 \)?
2. What is a geometric series, and how is it related to power series?
3. How does the differentiation of a geometric series help in solving power series involving \( n^2 \)?
4. Are there other methods to compute such series without using standard results?
5. Can the same approach be used for sums involving higher powers of \( n \) (e.g., \( n^3 \))?

**Tip**: When dealing with series, recognizing patterns in summation formulas can significantly speed up the calculation!

La somme de la série \( \sum_{n \geq 0} \frac{n^2}{2^n} \) est :

Discover the step-by-step solution to the infinite series \( \sum_{n \geq 0} \frac{n^2}{2^n} \). This detailed guide explores power series concepts and uses the known formula for sums involving quadratic terms. Suitable for university-level mathematics.

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