https://r2.erweima.ai/mathbot/image/17249264238778hp7h4zw.jpg

The series shown in the image is:

\[
\frac{1}{2} - \frac{1}{4} + \frac{1}{8} - \frac{1}{16} + \cdots
\]

This is an alternating geometric series where the first term \( a_1 \) is \( \frac{1}{2} \), and the common ratio \( r \) is \( -\frac{1}{2} \).

The general formula for the \( n \)th term of a geometric series is given by:

\[
a_n = a_1 \cdot r^{n-1}
\]

For this series:

\[
a_n = \frac{1}{2} \cdot \left(-\frac{1}{2}\right)^{n-1}
\]

So, the \( n \)th term is:

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

This formula represents the general term of the series.

Would you like further details on how this formula is derived or its applications? 

Here are five related questions you might find useful:

1. What is the sum of the infinite series?
2. How do you determine whether a geometric series converges?
3. Can this formula be applied to other types of series?
4. How does the alternating sign affect the convergence of the series?
5. What are the real-world applications of geometric series?

**Tip:** In alternating series like this one, the sign alternates between positive and negative, which often affects the series' convergence properties.

Explore the alternating geometric series with a focus on its general term formula. Learn how to derive and apply the formula \( a_n = \frac{(-1)^{n-1}}{2^n} \), and discover related concepts such as convergence properties and real-world applications.

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