The problem in the image appears to involve a series: $4 + 2 + 1 + \dots + \frac{1}{16}$. This is a geometric series with the first term $a = 4$ and a common ratio $r = \frac{1}{2}$ (each term is half the previous one).

Formula for the sum of a geometric series:

For a geometric series, the sum of the first $n$ terms is: $S_n = a \frac{1 - r^n}{1 - r}, \quad \text{if } r \neq 1$ Where:

• $a$ is the first term,
• $r$ is the common ratio,
• $n$ is the number of terms.

If the series continues indefinitely ($n \to \infty$), the sum of an infinite geometric series is: $S = \frac{a}{1 - r}, \quad \text{if } |r| < 1$

Solution:

Here, $a = 4$, $r = \frac{1}{2}$, and $|r| < 1$, so the series converges to: $S = \frac{4}{1 - \frac{1}{2}} = \frac{4}{\frac{1}{2}} = 8$

Thus, the sum of the series is: $S = 8$

Would you like a detailed breakdown or more examples?

Related Questions:

1. What is the formula for the $n$-th term of a geometric series?
2. How does the sum of an infinite geometric series depend on $r$?
3. What happens if $|r| \geq 1$ in a geometric series?
4. Can geometric series be used to approximate functions in mathematics?
5. How does the sum of a finite geometric series differ from an infinite one?

Tip:

For any infinite geometric series, always check if $|r| < 1$. If not, the series diverges and does not have a finite sum!