To find the sum to infinity of a geometric series, we use the formula:

$S_\infty = \frac{a}{1 - r}$

where:

• $a$ is the first term,
• $r$ is the common ratio, with $|r| < 1$ for the sum to converge.

In this series:

• The first term $a = 32$,
• The common ratio $r = \frac{16}{32} = \frac{1}{2}$.

Now we can substitute these values into the formula:

$S_\infty = \frac{32}{1 - \frac{1}{2}} = \frac{32}{\frac{1}{2}} = 32 \times 2 = 64$

Answer:

The sum to infinity of the series is $64$.

Would you like a detailed breakdown of these steps or have any questions?

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1. How do we derive the formula for the sum of an infinite geometric series?
2. Why must the common ratio $r$ be less than 1 in magnitude for convergence?
3. How would we calculate the sum if the series was finite?
4. Can a geometric series with a common ratio greater than 1 have a sum to infinity?
5. What happens to the sum if the common ratio $r$ is negative?

Tip: In a geometric series, if the common ratio is a fraction less than 1, the series will converge, making the infinite sum possible.