Problem 3 Analyzing Infinite Geometric Series

Does the series converge or diverge? If it converges, what is the sum?

1. 1 + 1 2 + 1 4 + … 1 plus , 1 half , plus , 1 fourth , plus dot dot dot

   Think

   When does an infinite geometric series converge?

   An infinite geometric series converges when the absolute value of the common ratio is less than 1.

   r = 1 2 ÷ 1 = 1 2 r equals , 1 half , divides 1 equals , 1 half

   Since | r | = | 1 2 | < 1 , absolute value of r , , equals absolute value of , 1 half , end absolute value , . less than 1 . comma  the series converges.

   S = a 1 1 − r = 1 1 − 1 2 = 1 1 2 = 2 s equals . fraction eh sub 1 , over 1 minus r end fraction . equals . fraction 1 , over 1 minus , 1 half end fraction . equals , fraction 1 , over and 1 half end fraction , equals 2

2. ∑ n = 0 ∞ ( 2 3 ) ( − 5 4 ) n sum , from , n equals 0 , to , infinity , of . open , 2 thirds , close . open , negative , 5 fourths , close to the n

   Since | r | = | − 5 4 | = 5 4 > 1 , absolute value of r , , equals absolute value of , negative , 5 fourths , end absolute value , . equals , 5 fourths , greater than 1 . comma  the series diverges.

✓ Got It?

3. Does the infinite series converge or diverge? If it converges, what is the sum?
   1. 1 2 + 3 4 + 9 8 + … 1 half , plus , 3 fourths , plus , 9 eighths , plus dot dot dot
   2. 1 3 − 1 9 + 1 27 − 1 81 + … 1 third , minus , 1 ninth , plus , 1 twenty seventh , minus , 1 eighty first , plus dot dot dot
   3. ∑ n = 1 ∞ ( 2 3 ) n sum , from , n equals 1 , to , infinity , of . open , 2 thirds , close to the n
   4. Reasoning Will an infinite geometric series either converge or diverge? Explain.

✓ Lesson Check

Do you know HOW?

Evaluate each finite geometric series.

1. 1 5 + 1 10 + 1 20 + 1 40 + 1 80 1 fifth , plus , 1 tenth , plus , 1 twentieth , plus , 1 fortieth , plus , 1 eightieth
2. 9 − 6 + 4 − 8 3 + 16 9 9 minus 6 plus 4 minus , 8 thirds , plus , 16 over 9

Determine whether each infinite geometric series diverges or converges.

3. 1 − 1 6 + 1 36 − 1 216 + … 1 minus , 1 sixth , plus , 1 thirty sixth , minus , 1 over 216 , plus dot dot dot
4. 1 64 + 1 32 + 1 16 + … 1 sixty fourth , plus , 1 thirty second , plus , 1 sixteenth , plus dot dot dot

Do you UNDERSTAND?

5. Error Analysis A classmate uses the formula for the sum of an infinite geometric series to evaluate 1 + 1.1 + 1.21 + 1.331 + … 1 plus 1.1 plus , 1.21 , plus , 1.331 , plus dot dot dot  and gets − 10 . negative 10 .  What error did your classmate make?
6. Writing Explain how you can determine whether an infinite geometric series has a sum.
7. Compare and Contrast How are the formulas for the sum of a finite arithmetic series and the sum of a finite geometric series similar? How are they different?