Find the Sum of an Infinite Geometric Series

Step-by-step solution

1. Find the first term and the common ratio.

   $$a = 8,\quad r = \frac{4}{8} = \frac{1}{2}$$

   Why: Each term is half of the one before it, so the ratio is 1/2.

2. Check that the series converges.

   $$|r| = \frac{1}{2} < 1$$

   Why: An infinite geometric series has a finite sum only when the ratio is between -1 and 1.

3. Use the sum formula.

   $$S = \frac{a}{1 - r} = \frac{8}{1 - 1/2} = \frac{8}{1/2} = 16$$

   Why: Dividing by 1/2 is the same as multiplying by 2.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

1. 1

   $1 + \frac{1}{2} + \frac{1}{4} + \cdots$

   Show answer

   $2$

2. 2

   $6 + 3 + \frac{3}{2} + \cdots$

   Show answer

   $12$

3. 3

   $9 + 3 + 1 + \cdots$

   Show answer

   $\frac{27}{2}$

4. 4

   $4 + 2 + 1 + \cdots$

   Show answer

   $8$

5. 5

   $2 + \frac{1}{2} + \frac{1}{8} + \cdots$

   Show answer

   $\frac{8}{3}$