Summation Calculator
Summation Calculator
Calculate the sum of mathematical expressions over a range
Use 'n' as the variable. Supported: +, -, *, /, ^(power), sqrt(), sin(), cos(), tan(), log(), ln(), pi, e
Summation Calculator: Complete Mathematical Guide
Summation (Σ notation) represents the sum of a sequence of terms following a specific pattern.This mathematical notation allows us to express long sums concisely and is fundamental in calculus, statistics, and many areas of mathematics and science.
Our professional summation calculator evaluates mathematical expressions over specified ranges, supporting complex functions, trigonometric operations, and advanced mathematical notation. Perfect for students, researchers, and professionals working with series and sequences.
Quick Answer
To calculate a summation: Enter your expression using 'n' as the variable, set the lower and upper bounds, and this calculator will evaluate each term and provide the total sum. For example, Σ(n²) from n=1 to 5 calculates 1² + 2² + 3² + 4² + 5² = 55.
Mathematical Foundation
Summation from n=a to n=b, where f(n) is the expression evaluated at each value of n
Key Summation Concepts:
Sigma Notation (Σ)
The Greek letter sigma (Σ) represents summation. The expression below sigma shows what to sum, while the range (from lower bound to upper bound) indicates which values to substitute.
Index Variable
The variable 'n' (or any letter) serves as the index, taking on each integer value from the lower bound to the upper bound. Each substitution creates a term in the sum.
Finite vs Infinite Series
Finite summations have specific start and end points, while infinite series continue indefinitely. This calculator handles finite sums with practical upper limits.
Common Mathematical Series
Arithmetic Series
Σ(a + (n-1)d) = na + n(n-1)d/2Geometric Series
Σ(ar^n) = a(1-r^(n+1))/(1-r) for r ≠ 1Power Series
Σ(n^k) - Sum of k-th powersApplications of Summation
Mathematics & Science
Calculus Integration
Riemann sums approximate definite integrals using summation of rectangular areas
Statistics
Calculate means, variances, and other statistical measures using summation formulas
Physics
Work calculations, wave superposition, and quantum mechanical problems
Engineering & Technology
Digital Signal Processing
Fourier transforms and filter design using discrete summations
Computer Science
Algorithm analysis, complexity calculations, and discrete mathematics
Finance
Present value calculations, annuity payments, and compound interest
Example Problems with Solutions
Example 1: Sum of First n Natural Numbers
Calculate Σ(n) from n=1 to 100
Answer: 5,050
Example 2: Geometric Series
Calculate Σ(3·2^n) from n=0 to 5
Answer: 189
Example 3: Trigonometric Sum
Calculate Σ(sin(n·π/6)) from n=0 to 6
Answer: ≈ 3.732
Expression Input Guide
Supported Operations
Mathematical Constants
Important Notes
- • Use 'n' as the variable in your expression
- • Maximum range limited to 10,000 terms for performance
- • Complex expressions may take longer to compute
- • Check for mathematical validity (no division by zero, etc.)
- • Trigonometric functions use radians
Related Mathematical Tools
Summation Calculator
Calculate the sum of mathematical expressions over a range
Use 'n' as the variable. Supported: +, -, *, /, ^(power), sqrt(), sin(), cos(), tan(), log(), ln(), pi, e