The image contains a mathematical problem involving infinite series. Here’s a breakdown of the tasks:

1. Find the sum of the first $n$ terms of the series $S_n$.
2. Prove the convergence of the series using the definition of convergence.
3. Find the sum of the series $S$.

To address each of these points, I'll first transcribe and analyze the specific series from the image, then solve each part accordingly.

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Step 1: Transcription of the Series

The series are given as follows:

1. $S_n = 1 + \frac{1}{2} + \frac{1}{4} + \cdots + \frac{1}{2^{n-1}}$
2. $S_n = 1 - \frac{1}{2} + \frac{1}{4} - \cdots + (-1)^{n-1} \frac{1}{2^{n-1}}$
3. $S_n = 1 + \frac{1}{3} + \frac{1}{9} + \cdots + \frac{1}{3^{n-1}}$

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Step 2: Solving Each Part

1. Finding the sum of the first $n$ terms $S_n$:

1. For the first series:

   • This is a geometric series where the first term $a = 1$ and the common ratio $r = \frac{1}{2}$.
   • The sum of the first $n$ terms is: $S_n = \frac{1 \cdot (1 - \left(\frac{1}{2}\right)^n)}{1 - \frac{1}{2}} = 2 \left(1 - \frac{1}{2^n}\right)$

2. For the second series:

   • This is an alternating geometric series with first term $a = 1$ and common ratio $r = -\frac{1}{2}$.
   • The sum of the first $n$ terms is: $S_n = \frac{1 \cdot (1 - (-\frac{1}{2})^n)}{1 - (-\frac{1}{2})} = \frac{2}{3} \left(1 - (-\frac{1}{2})^n\right)$

3. For the third series:

   • This is also a geometric series with first term $a = 1$ and common ratio $r = \frac{1}{3}$.
   • The sum of the first $n$ terms is: $S_n = \frac{1 \cdot (1 - \left(\frac{1}{3}\right)^n)}{1 - \frac{1}{3}} = \frac{3}{2} \left(1 - \frac{1}{3^n}\right)$

2. Proving the Convergence:

To prove convergence, we examine the limit of the sum $S_n$ as $n$ approaches infinity.

• For the first series: $\lim_{n \to \infty} S_n = \lim_{n \to \infty} 2 \left(1 - \frac{1}{2^n}\right) = 2 \times 1 = 2$ The series converges to 2.

• For the second series: $\lim_{n \to \infty} S_n = \lim_{n \to \infty} \frac{2}{3} \left(1 - (-\frac{1}{2})^n\right) = \frac{2}{3} \times 1 = \frac{2}{3}$ The series converges to $\frac{2}{3}$.

• For the third series: $\lim_{n \to \infty} S_n = \lim_{n \to \infty} \frac{3}{2} \left(1 - \frac{1}{3^n}\right) = \frac{3}{2} \times 1 = \frac{3}{2}$ The series converges to $\frac{3}{2}$.

3. Finding the sum of the series $S$:

Given the convergence proof, the sums of the series are:

1. Sum of the first series: $S = 2$
2. Sum of the second series: $S = \frac{2}{3}$
3. Sum of the third series: $S = \frac{3}{2}$

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Would you like a detailed explanation of any part of this process? Here are five related questions to deepen your understanding:

1. How do you determine the convergence of a geometric series?
2. Why does the alternating series converge more slowly?
3. What is the significance of the common ratio in determining the sum of a series?
4. Can the method used here be applied to non-geometric series?
5. How would the convergence proof differ for a divergent series?

Tip: Always check the common ratio in a geometric series; if the absolute value of the common ratio is less than 1, the series will converge.