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To solve these geometric sequences and find their sum to infinity, we use the formula for the sum of an infinite geometric series:

\[
S_{\infty} = \frac{a}{1 - r}
\]
where:
- \( a \) is the first term,
- \( r \) is the common ratio (the factor between successive terms), and
- The series converges if \( |r| < 1 \).

Let's solve each of the sequences provided.

### 1. Sequence: \( 64, 16, 4, 1, \dots \)
- First term \( a = 64 \).
- Common ratio \( r = \frac{16}{64} = \frac{1}{4} \).
- Since \( |r| < 1 \), the sum to infinity exists:
  
  \[
  S_{\infty} = \frac{64}{1 - \frac{1}{4}} = \frac{64}{\frac{3}{4}} = 64 \times \frac{4}{3} = \frac{256}{3}
  \]

### 2. Sequence: \( 3, 1, \frac{1}{3}, \frac{1}{9}, \dots \)
- First term \( a = 3 \).
- Common ratio \( r = \frac{1}{3} \).
- Since \( |r| < 1 \), the sum to infinity exists:
  
  \[
  S_{\infty} = \frac{3}{1 - \frac{1}{3}} = \frac{3}{\frac{2}{3}} = 3 \times \frac{3}{2} = \frac{9}{2}
  \]

### 3. Sequence: \( -4, 1, -\frac{1}{4}, \frac{1}{16}, \dots \)
- First term \( a = -4 \).
- Common ratio \( r = -\frac{1}{4} \).
- Since \( |r| < 1 \), the sum to infinity exists:
  
  \[
  S_{\infty} = \frac{-4}{1 - \left( -\frac{1}{4} \right)} = \frac{-4}{1 + \frac{1}{4}} = \frac{-4}{\frac{5}{4}} = -4 \times \frac{4}{5} = -\frac{16}{5}
  \]

### 4. Sequence: \( 24, 4, \frac{2}{3}, \dots \)
- First term \( a = 24 \).
- Common ratio \( r = \frac{4}{24} = \frac{1}{6} \).
- Since \( |r| < 1 \), the sum to infinity exists:
  
  \[
  S_{\infty} = \frac{24}{1 - \frac{1}{6}} = \frac{24}{\frac{5}{6}} = 24 \times \frac{6}{5} = \frac{144}{5}
  \]

### 5. Sequence: \( 1, \sqrt{2}, 2, 2\sqrt{2}, \dots \)
- First term \( a = 1 \).
- Common ratio \( r = \frac{\sqrt{2}}{1} = \sqrt{2} \).
- Here, \( |r| = \sqrt{2} > 1 \), so the sum to infinity **does not exist** because the series diverges.

### Summary:
1. \( S_{\infty} = \frac{256}{3} \)
2. \( S_{\infty} = \frac{9}{2} \)
3. \( S_{\infty} = -\frac{16}{5} \)
4. \( S_{\infty} = \frac{144}{5} \)
5. Sum does not exist (diverges).

Would you like further details on any of these calculations or have any other questions?

### Related questions:
1. How do you determine the common ratio in a geometric sequence?
2. What are the conditions for a geometric series to have a sum to infinity?
3. How can you prove that a geometric series diverges when \( |r| \geq 1 \)?
4. What is the formula for the sum of a finite geometric series?
5. Can a geometric sequence have negative terms and still converge?

**Tip:** Always check if the absolute value of the common ratio \( r \) is less than 1 to ensure that the sum to infinity exists!

Find the sum to infinity of each geometric sequence, if it exists.

Explore the sum to infinity of geometric sequences with a step-by-step solution. Learn the conditions for convergence, the formula for the sum to infinity, and analyze different sequences to determine if they converge or diverge.

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