Thank you for the clarification. Let's solve the problem with the correct expression for \( a_n \), which is given as:

\[
a_n = \frac{1}{2^{2n}}
\]

### Step 1: Analyze the Sequence \( a_n \)
Given \( a_n = \frac{1}{2^{2n}} \), we can rewrite this as:

\[
a_n = \frac{1}{(2^2)^n} = \frac{1}{4^n}
\]

### Step 2: Analyze the Sum \( S_n \)
The sum \( S_n \) is defined as the sum of the first \( n \) terms of the sequence:

\[
S_n = \sum_{k=0}^{n-1} a_k = \sum_{k=0}^{n-1} \frac{1}{4^k}
\]

### Step 3: Recognize the Sum as a Geometric Series
This sum \( S_n \) is a geometric series with the first term \( a_0 = 1 \) (since \( 4^0 = 1 \)) and the common ratio \( r = \frac{1}{4} \). The sum of the first \( n \) terms of a geometric series is given by:

\[
S_n = \frac{1-r^n}{1-r}
\]

Substituting \( r = \frac{1}{4} \):

\[
S_n = \frac{1 - \left(\frac{1}{4}\right)^n}{1 - \frac{1}{4}} = \frac{1 - \frac{1}{4^n}}{\frac{3}{4}} = \frac{4}{3} \left(1 - \frac{1}{4^n}\right)
\]

### Step 4: Take the Limit as \( n \to \infty \)
Now, take the limit of \( S_n \) as \( n \to \infty \):

\[
\lim_{n \to \infty} S_n = \lim_{n \to \infty} \frac{4}{3} \left(1 - \frac{1}{4^n}\right)
\]

As \( n \to \infty \), \( \frac{1}{4^n} \) approaches 0. Therefore:

\[
\lim_{n \to \infty} S_n = \frac{4}{3} \times 1 = \frac{4}{3}
\]

### Final Answer
The limit of \( S_n \) as \( n \to \infty \) is \( \boxed{\frac{4}{3}} \).

Would you like further details or have any questions? Here are some related questions that might interest you:

1. What are the conditions for the convergence of a geometric series?
2. How would the result change if the common ratio were greater than 1?
3. Can you find the sum of an infinite geometric series when \( r < 1 \)?
4. How does the initial term affect the sum of a geometric series?
5. What would happen if \( a_n \) was defined as \( \frac{1}{2^{n^2}} \) instead?

**Tip:** For any geometric series, always verify the common ratio to determine convergence before summing the series.

Explore how to solve the limit of the sum of a geometric series as n approaches infinity with a detailed step-by-step solution. This problem covers key mathematical concepts including geometric series, limits, and convergence theorems. Suitable for advanced high school students and anyone interested in advanced mathematics.

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