Sum of Series Calculator
Calculate the sum of arithmetic, geometric, and infinite series with step-by-step solutions
Starting value of the series
Difference between consecutive terms
Must be |r| < 1 for convergence
Ratio between consecutive terms
How many terms to sum
Quick Examples
Sum of Arithmetic Series (S) Sum of Geometric Series (S) Sum of Infinite Series (S∞)
Sum of terms Infinite sum converges to this value
First 15 Terms Sequence Terms
Step-by-Step Calculation
Series Sum
S∞ S
First Term
a₁
Common Difference Common Ratio
d r
Series Formulas Reference
| Series Type | Sum Formula | Condition |
|---|---|---|
| Arithmetic | Sₙ = n/2 × [2a + (n-1)d] | n ≥ 1 |
| Geometric (r ≠ 1) | Sₙ = a(1 - rⁿ)/(1 - r) | r ≠ 1 |
| Geometric (r = 1) | Sₙ = n × a | r = 1 |
| Infinite Geometric | S∞ = a/(1 - r) | |r| < 1 |
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About Sum of Series Calculator
What is a Sum of Series Calculator?
A sum of series calculator is a mathematical tool that computes the total of all terms in a sequence. This calculator supports three types of series: arithmetic series (constant difference between terms), geometric series (constant ratio between terms), and infinite geometric series (convergent series with infinite terms).
How to Use This Calculator
- Select Series Type: Choose arithmetic, geometric, or infinite geometric series
- Enter Values: Input the first term, common difference/ratio, and number of terms
- View Results: Get instant sum calculation with step-by-step breakdown
- Explore Terms: See the generated sequence terms
Understanding Series Types
Arithmetic Series
An arithmetic series has a constant difference between consecutive terms.
- Formula: Sₙ = n/2 × (2a + (n-1)d) or Sₙ = n/2 × (a₁ + aₙ)
- Example: 2 + 5 + 8 + 11 + 14 (d = 3)
Geometric Series
A geometric series has a constant ratio between consecutive terms.
- Formula: Sₙ = a(1 - rⁿ)/(1 - r) for r ≠ 1
- Example: 2 + 6 + 18 + 54 (r = 3)
Infinite Geometric Series
An infinite geometric series converges when |r| < 1.
- Formula: S∞ = a/(1 - r) for |r| < 1
- Example: 1 + 0.5 + 0.25 + 0.125 + ... = 2
The Formulas Behind the Calculations
Arithmetic Series Sum
Formula: Sₙ = n/2 × [2a + (n - 1)d]
Where:
- Sₙ = Sum of first n terms
- a = First term
- n = Number of terms
- d = Common difference
Geometric Series Sum (Finite)
Formula: Sₙ = a(1 - rⁿ)/(1 - r) for r ≠ 1
Where:
- Sₙ = Sum of first n terms
- a = First term
- r = Common ratio
- n = Number of terms
Infinite Geometric Series Sum
Formula: S∞ = a/(1 - r) for |r| < 1
The series only converges when the absolute value of the ratio is less than 1.
Frequently Asked Questions
What is the difference between a sequence and a series?
A sequence is an ordered list of numbers, while a series is the sum of the terms in a sequence.
When does an infinite geometric series converge?
An infinite geometric series converges (has a finite sum) only when the absolute value of the common ratio is less than 1 (|r| < 1).
Can I calculate the sum of any series?
This calculator handles arithmetic and geometric series. Other types like harmonic, telescoping, or power series require different approaches.
What happens when r = 1 in a geometric series?
When r = 1, each term equals the first term, so Sₙ = n × a (simply multiply the first term by the number of terms).
Important Limitations
- For infinite geometric series, |r| must be less than 1 for convergence
- Very large values of n may result in computational overflow
- This calculator does not handle alternating series or complex ratios
Note: This calculator is for educational purposes. Always verify critical calculations manually.