Geometric Series Calculator
Geometric Series Calculator
Calculate sum of finite and infinite geometric series. Check convergence, find common ratio, and get step-by-step solutions.
Geometric Series Result
Sum = 1.999...
Series Analysis:
Series Visualization:
Step-by-Step Calculation:
Mathematical Analysis:
Geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
What is a Geometric Series?
Geometric series is a series of the form a + ar + ar² + ar³ + ... where 'a' is the first term and 'r' is the common ratio. The sum of the first n terms is given by Sₙ = a(1 - rⁿ)/(1 - r) for r ≠ 1. For |r| < 1, the infinite geometric series converges to S = a/(1 - r).
Types of Geometric Series
Convergent Series
Sum approaches finite limit
Example: 1 + ½ + ¼ + ... = 2
Divergent Series
Sum grows without bound
Example: 1 + 2 + 4 + ... → ∞
Alternating Series
Terms alternate signs
Example: 1 - ½ + ¼ - ...
Finite Series
Exact sum calculation
Example: 1 + 2 + 4 + 8 + 16
Geometric Series Formulas
1. Finite Geometric Series Sum
Sₙ = a(1 - rⁿ)/(1 - r) (for r ≠ 1)
Sₙ = na (for r = 1)
2. Infinite Geometric Series Sum
S = a/(1 - r) (for |r| < 1)
Diverges for |r| ≥ 1
3. nth Term Formula
aₙ = arⁿ⁻¹
Convergence Conditions
| Condition on r | Series Behavior | Sum Formula | Example |
|---|---|---|---|
| |r| < 1 | Converges absolutely | S = a/(1 - r) | 1 + ½ + ¼ + ... = 2 |
| r = 1 | Diverges (unless a=0) | Sₙ = na → ∞ | 1 + 1 + 1 + ... → ∞ |
| r = -1 | Oscillates, diverges | No limit | 1 - 1 + 1 - 1 + ... oscillates |
| |r| > 1 | Diverges to ±∞ | Sₙ → ±∞ | 1 + 2 + 4 + ... → ∞ |
| -1 < r < 0 | Converges (alternating) | S = a/(1 - r) | 1 - ½ + ¼ - ... = ⅔ |
Real-World Applications
Finance & Economics
- Compound interest: Future value calculations
- Annuities: Regular payment valuations
- Depreciation: Declining balance method
- Economic growth: Multiplier effect in economics
Physics & Engineering
- Radioactive decay: Half-life calculations
- Circuit analysis: Impedance in AC circuits
- Optics: Multiple reflections in mirrors
- Mechanical systems: Damped oscillations
Computer Science
- Algorithm analysis: Time complexity of divide-and-conquer
- Data compression: Geometric distributions
- Network theory: Propagation delays
- Game theory: Repeated games with discounting
Biology & Medicine
- Population growth: Geometric growth models
- Drug dosage: Repeated medication administration
- Epidemiology: Spread of diseases
- Genetics: Probability in inheritance
Common Geometric Series Examples
| Series | First Term (a) | Ratio (r) | Sum (Finite n=5) | Sum (Infinite) |
|---|---|---|---|---|
| 1 + 2 + 4 + 8 + ... | 1 | 2 | 31 | ∞ (diverges) |
| 1 + ½ + ¼ + ⅛ + ... | 1 | ½ | 1.9375 | 2 |
| 3 + 1 + ⅓ + ⅑ + ... | 3 | ⅓ | 4.4815 | 4.5 |
| 1 - ½ + ¼ - ⅛ + ... | 1 | -½ | 0.6875 | ⅔ ≈ 0.6667 |
| 0.9 + 0.09 + 0.009 + ... | 0.9 | 0.1 | 0.99999 | 1 |
Step-by-Step Calculation Examples
Example 1: Finite Series 2 + 6 + 18 + 54 (n=4)
- Identify first term: a = 2
- Find common ratio: r = 6/2 = 3
- Number of terms: n = 4
- Apply formula: Sₙ = a(1 - rⁿ)/(1 - r)
- Calculate: S₄ = 2(1 - 3⁴)/(1 - 3)
- Simplify: 2(1 - 81)/(-2) = 2(-80)/(-2) = 80
- Verify: 2 + 6 + 18 + 54 = 80 ✓
Example 2: Infinite Series 1 + ⅓ + ⅑ + ...
- Identify first term: a = 1
- Find common ratio: r = (⅓)/1 = ⅓
- Check convergence: |r| = |⅓| = 0.333 < 1 ✓
- Apply formula: S = a/(1 - r)
- Calculate: S = 1/(1 - ⅓) = 1/(⅔) = 3/2 = 1.5
- Partial sums verification:
- S₁ = 1
- S₂ = 1 + ⅓ ≈ 1.333
- S₃ = 1 + ⅓ + ⅑ ≈ 1.444
- S₄ = 1 + ⅓ + ⅑ + 1/27 ≈ 1.481
- Approaching 1.5 ✓
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Frequently Asked Questions (FAQs)
Q: What's the difference between geometric sequence and geometric series?
A: Geometric sequence is the list of terms: a, ar, ar², ar³, ... Geometric series is the sum of these terms: a + ar + ar² + ar³ + ...
Q: When does an infinite geometric series converge?
A: An infinite geometric series converges if and only if |r| < 1. The sum is then a/(1 - r). If |r| ≥ 1, the series diverges.
Q: Can a geometric series have a negative common ratio?
A: Yes! If -1 < r < 0, the series converges (alternating signs). If r ≤ -1, the series diverges or oscillates.
Q: How is geometric series used in compound interest?
A: Compound interest formula A = P(1 + r)ⁿ is essentially a geometric sequence. The sum of regular investments forms a geometric series.
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