33. Consider the following infinite series:
1 + r + r2 + r3 + … + ∞
If r = 0.3, then the sum of this infinite series is __________.
This converges because |r| < 1. The formula S = a/1−r applies where a=1 is the first term.
Derivation
Let S = 1 + 0.3 + 0.32 + 0.33 + …
Multiply by r: 0.3S = 0.3 + 0.32 + 0.33 + …
Subtract: S − 0.3S = 1, so 0.7S = 1, and S = 1/0.7 = 10/7
No options are provided in the query, so none require separate evaluation; the exact sum fills the blank.
Formula Explained
The infinite geometric series formula is S = 1/1−r for first term 1 and |r| < 1. Here, S = 1/1−0.3 = 10/7.
- Derivation: S = 1+r+r2+…, rS = r+r2+…, S(1−r) = 1
- Convergence requires |r| < 1; otherwise, it diverges.
Step-by-Step Calculation
For r=0.3:
- Identify a=1, r=0.3
- Check |0.3| < 1—valid
- S = 10/7 = 1.428571‾
Partial sums approximate: 1 = 1, 1+0.3 = 1.3, 1.3+0.09 = 1.39, approaching 10/7.
Applications
Used in finance (perpetuities), physics (oscillations), and engineering. For biotech modeling like microbial growth decay (r < 1), it predicts limits.
