Infinite Series

Step 1: Identify a₁ and r: a₁ = 1 r = (1/2)/1 = 1/2

Step 2: Check convergence condition: For convergence: |r| < 1 |1/2| = 1/2 < 1 ✓

Step 3: The series CONVERGES

Step 4: Find the sum: S = a₁/(1 - r) S = 1/(1 - 1/2) S = 1/(1/2) S = 2

Answer: Converges to 2

Step 1: Identify the series type: This is geometric with a₁ = 2, r = 2

Step 2: Check convergence condition: For convergence: |r| < 1 |2| = 2 ≥ 1 ✗

Step 3: The series DIVERGES

Step 4: Explain why: The terms keep getting larger (2, 4, 8, 16, ...) They don't approach zero, so the sum grows without bound

Answer: Diverges (no finite sum)

Step 1: Identify a₁ and r: a₁ = 9 r = 3/9 = 1/3

Step 2: Verify it's geometric: 1/3 = (1/3)/1 ✓ Ratio is consistent

Step 3: Check convergence: |1/3| < 1 ✓ Series converges

Step 4: Use infinite sum formula: S = a₁/(1 - r) S = 9/(1 - 1/3) S = 9/(2/3) S = 9 × 3/2 S = 27/2 S = 13.5

Answer: 27/2 or 13.5

Step 1: Write as a series: 0.272727... = 0.27 + 0.0027 + 0.000027 + ... = 27/100 + 27/10000 + 27/1000000 + ...

Step 2: Factor out 27/100: = (27/100)(1 + 1/100 + 1/10000 + ...)

Step 3: Identify the geometric series: Inside parentheses: a₁ = 1, r = 1/100

Step 4: Sum the geometric series: S = 1/(1 - 1/100) S = 1/(99/100) S = 100/99

Step 5: Multiply: 0.272727... = (27/100) × (100/99) = 27/99

Step 6: Simplify: 27/99 = 3/11

Step 7: Verify: 3 ÷ 11 = 0.272727... ✓

Answer: 3/11

Step 1: Identify the geometric series: This is geometric with a₁ = (x/2)⁰ = 1 and r = x/2

Step 2: Determine convergence condition: Series converges when |r| < 1 |x/2| < 1

Step 3: Solve the inequality: |x/2| < 1 |x| < 2 -2 < x < 2

Step 4: Find the sum when it converges: S = a₁/(1 - r) S = 1/(1 - x/2) S = 1/[(2 - x)/2] S = 2/(2 - x)

Step 5: Verify at a specific value (e.g., x = 1): r = 1/2, |r| < 1 ✓ S = 2/(2 - 1) = 2 Check: 1 + 1/2 + 1/4 + ... = 2 ✓

Answer: Converges for -2 < x < 2 Sum = 2/(2 - x)