Calculate summation notation with step-by-step solutions. Perfect for arithmetic, geometric, polynomial, and factorial series in calculus and mathematics.
Enter bounds and expression to see summation notation
Comprehensive series calculations with detailed explanations
Calculate arithmetic series sums
Calculate geometric series sums
Calculate polynomial series sums
Calculate factorial series sums
Understanding series and summation notation
Input lower bound, upper bound, and expression.
Use appropriate summation formulas for the series type.
Receive sum, terms, and step-by-step solutions.
S = n(a₁ + aₙ)/2
S = a₁(rⁿ - 1)/(r - 1)
∑n² = n(n+1)(2n+1)/6
n! = n × (n-1) × ... × 1
Common series calculations and their results
Result: 15
Result: 30
Result: Average value calculation
Result: Future value calculation
Common questions about summation notation calculations
Summation notation (∑) is a mathematical notation used to represent the sum of a sequence of terms. It provides a compact way to write sums, especially when dealing with many terms or series.
Summation notation ∑(n=a to b) f(n) means "sum f(n) from n=a to n=b". The variable n is the index, a is the lower bound, b is the upper bound, and f(n) is the expression being summed.
An arithmetic series has a constant difference between consecutive terms (e.g., 2, 5, 8, 11...). A geometric series has a constant ratio between consecutive terms (e.g., 2, 4, 8, 16...).
Use the formula S = n(a₁ + aₙ)/2, where n is the number of terms, a₁ is the first term, and aₙ is the last term. This is much faster than adding all terms individually.
Use the formula S = a₁(rⁿ - 1)/(r - 1), where a₁ is the first term, r is the common ratio, and n is the number of terms. This formula works when r ≠ 1.
Partial sums are the sums of the first k terms of a series. They help visualize how the series sum grows and are useful for understanding convergence in infinite series.
Common formulas include: ∑n = n(n+1)/2, ∑n² = n(n+1)(2n+1)/6, ∑n³ = [n(n+1)/2]². These can be derived using mathematical induction or calculus.
Factorial sums involve calculating n! for each term and adding them together. Since factorials grow very rapidly, these sums are typically calculated directly rather than using closed-form formulas.
Summation notation is fundamental in calculus for defining Riemann sums, which lead to definite integrals. It's also used in Taylor series, Fourier series, and other infinite series expansions.
Summation notation is used in statistics (mean calculations), finance (compound interest), physics (momentum calculations), engineering (load distribution), and computer science (algorithm complexity analysis).
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