Q.30 The sum of the infinite geometric series 1 + 1/3 + 1/32 + 1/33 + …
(rounded off to one decimal place) is _________.
Sum of Infinite Geometric Series 1 + 1/3 + 1/32 + 1/33 + … Rounded to One Decimal Place
The series 1 + 1/3 + 1/32 + 1/33 + … is an infinite geometric series with first term a = 1 and common ratio r = 1/3. Its sum is 1.5 when rounded to one decimal place.
Series Identification
The given progression starts with 1, followed by 1/3, 1/32 (equivalent to 1/9), 1/33 (1/27), and continues indefinitely. Each term multiplies the previous by 1/3, confirming geometric nature since |r| = 1/3 < 1 ensures convergence.
Sum Calculation
Apply the infinite geometric series formula S = a / (1 – r), where a = 1 and r = 1/3. This yields:
S = 1 / (1 – 1/3) = 1 / (2/3) = 3/2 = 1.5 exactly. Rounding 1.5 to one decimal place remains 1.5.
Common Misconceptions
Interpretations like 1 + 1/3 + (1/3)2 + (1/3)3 match the formula directly, but errors arise from misreading as 1/32 or non-geometric. Alternate ratios like r = -1/3 yield S = 3/4 = 0.8, which diverges from the positive terms here.
Verification Steps
- First term: 1
- Ratio check: (1/3)/1 = 1/3; (1/9)/(1/3) = 1/3
- Partial sums: S3 ≈ 1 + 0.333 + 0.111 = 1.444; S4 ≈ 1.592, approaching 1.5
Such series appear in probability, finance, and physics for convergent processes.
