Calculate sigma notation sums with step-by-step solutions. Perfect for arithmetic, geometric, and polynomial series in calculus and mathematics.
Enter bounds and expression to see sigma notation
Comprehensive series calculations with detailed explanations
Calculate arithmetic series sums
Calculate geometric series sums
Calculate polynomial series sums
Calculate infinite series convergence
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
∑n³ = [n(n+1)/2]²
Common series calculations and their results
Result: 15
Result: 15
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Result: Total amplitude calculation
Common questions about sigma notation calculations
Sigma 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 infinite series.
Sigma 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.
Use convergence tests like the ratio test, comparison test, or integral test. For geometric series, convergence depends on the common ratio: |r| < 1 for convergence.
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.
Sigma 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.
Sigma notation is used in finance (compound interest), physics (wave superposition), statistics (mean calculations), engineering (load distribution), and computer science (algorithm complexity analysis).
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