Sum of Series Calculator
Calculate Mathematical Series with Formulas & Step-by-Step Solutions
Solve arithmetic series, geometric series, harmonic series, and custom mathematical expressions. Get instant results with detailed formulas and convergence analysis.
Supported Series Types
Predefined Series
- • Arithmetic Series: a + (a+d) + (a+2d) + ...
- • Geometric Series: a + ar + ar² + ar³ + ...
- • Square Numbers: 1² + 2² + 3² + ... = n(n+1)(2n+1)/6
- • Cube Numbers: 1³ + 2³ + 3³ + ... = [n(n+1)/2]²
- • Harmonic Series: 1 + 1/2 + 1/3 + 1/4 + ...
- • Factorial Series: 1! + 2! + 3! + ... + n!
- • Fibonacci Series: F₁ + F₂ + F₃ + ... + Fₙ
Custom Expressions
- • Basic operations: +, -, *, /, ^ (power)
- • Trigonometric: sin(x), cos(x), tan(x)
- • Logarithmic: log(x), ln(x)
- • Other functions: sqrt(x), abs(x)
- • Factorial notation: n!
- • Constants: pi (π), e
- • Examples: n², 2n+1, sin(nπ/6), n!/2ⁿ
Common Series Formulas
Arithmetic Series
Sₙ = n/2 × (2a + (n-1)d)
where a = first term, d = common difference, n = number of terms
Geometric Series
Sₙ = a(1-rⁿ)/(1-r)
where a = first term, r = common ratio (r ≠ 1)
Sum of Squares
Σn² = n(n+1)(2n+1)/6
Sum of first n square numbers
Sum of Cubes
Σn³ = [n(n+1)/2]²
Sum of first n cube numbers
Infinite Geometric
S∞ = a/(1-r)
when |r| < 1 (converges)
Harmonic Numbers
Hₙ ≈ ln(n) + γ
where γ ≈ 0.5772 (Euler-Mascheroni constant)
Frequently Asked Questions
How do I calculate an arithmetic series?
Enter the first term (a), common difference (d), and number of terms (n). The calculator uses the formula Sₙ = n/2 × (2a + (n-1)d) to find the sum instantly.
What's the difference between finite and infinite series?
Finite series have a specific number of terms and always have a definite sum. Infinite series continue forever and may converge to a finite value or diverge. Our calculator handles finite series and provides convergence information for applicable infinite series.
Can I create custom mathematical expressions?
Yes! Use the "Custom Expression" tab to enter any mathematical formula using variables like n, x, or k. Supports functions like sin(n), n², factorial (n!), and constants like π and e.
What's the maximum number of terms I can calculate?
The calculator supports up to 1,000 terms for most series types. Factorial series are limited to 20 terms to prevent numerical overflow. This covers virtually all practical mathematical applications.
How accurate are the calculations?
All calculations use exact formulas when available (like arithmetic and geometric series) or high-precision floating-point arithmetic. Results are accurate to at least 10 decimal places for most practical purposes.
Can I export or save my results?
Yes! You can copy individual results to your clipboard or download the complete calculation breakdown as a text file, including the series type, formula, sum, and detailed term-by-term calculations.
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Learn more about calculations with our helpful tutorials and guides.
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