https://file.aiquickdraw.com/mathbot/file/1727399160337pgrilmbs.jpg

Let's break down the problem displayed in the image.

### Part c:
The formula given for the sum of the series is:

\[
S_n = 2 \cdot (-1/2)^{n-1}
\]

This is an explicit formula for the sum of a finite series up to \(n\) terms.

### Part d:
You are asked to find the sum of the **infinite series**:

\[
S = 2 - 1 + \frac{1}{2} - \frac{1}{4} + \cdots + 2^{1-n} \cdot (-1)^n + \cdots
\]

This looks like an infinite geometric series. The sum of an infinite geometric series \(S_{\infty}\) is given by:

\[
S_{\infty} = \frac{a}{1 - r}
\]

where \(a\) is the first term of the series and \(r\) is the common ratio. In this case:
- The first term \(a = 2\).
- The common ratio \(r = -\frac{1}{2}\).

Thus, applying the formula:

\[
S_{\infty} = \frac{2}{1 - (-1/2)} = \frac{2}{1 + 1/2} = \frac{2}{3/2} = \frac{4}{3}
\]

So the sum of the infinite series is:

\[
S = \frac{4}{3}
\]

### Follow-up questions:
1. What is the general formula for the sum of a finite geometric series?
2. How does the value of \(r\) affect whether a geometric series converges or diverges?
3. Can this formula be used for any geometric series, or are there specific conditions?
4. What happens to the sum of the series when \(n\) approaches infinity for a divergent series?
5. How do you differentiate between a geometric and an arithmetic series?

### Tip:
Always check the common ratio in a geometric series; if \(|r| \geq 1\), the series will not converge.

In terms of n, find an explicit formula for the sum of the series S_n. Then, find the sum of the infinite series as n approaches infinity.

Learn how to find the explicit formula for the sum of a geometric series and calculate its infinite sum. This problem applies geometric series formulas and covers the concepts of convergence. Suitable for advanced high school math students.

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