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.