Struct SummationMethods 

pub struct SummationMethods;

Summation methods and utilities

Performance

All methods use closed-form formulas when applicable, providing O(1) time complexity for arithmetic/geometric series and power sums (k ≤ 3). General sums remain symbolic.

Implementations

impl SummationMethods

pub fn arithmetic_series( first_term: &Expression, common_difference: &Expression, num_terms: &Expression, ) -> Expression

Compute arithmetic series sum: Σ(a + (i-1)d) from i=1 to n

Formula

Sum = n/2 * (2a + (n-1)d)

Performance

• Time Complexity: O(1) - uses closed-form formula
• Space Complexity: O(1) - constant expression construction

pub fn geometric_series( first_term: &Expression, common_ratio: &Expression, num_terms: &Expression, ) -> Expression

Compute geometric series sum: Σ(ar^(i-1)) from i=1 to n

Formula

Sum = a * (1 - r^n) / (1 - r) for r ≠ 1

Performance

• Time Complexity: O(1) - uses closed-form formula
• Space Complexity: O(1) - constant expression construction

pub fn infinite_geometric_series( first_term: &Expression, common_ratio: &Expression, ) -> Expression

Compute infinite geometric series sum: Σ(ar^(i-1)) from i=1 to ∞

Formula

Sum = a / (1 - r) for |r| < 1

Domain Restriction

Convergence requires |r| < 1. For |r| ≥ 1, series diverges.

Performance

• Time Complexity: O(1) - uses closed-form formula
• Space Complexity: O(1) - constant expression construction

pub fn power_sum(power: &Expression, upper_limit: &Expression) -> Expression

Compute power sum: Σ(i^k) from i=1 to n using Faulhaber’s formulas

Formulas (SymPy Validated)

• k=0: Σ1 = n
• k=1: Σi = n(n+1)/2
• k=2: Σi² = n(n+1)(2n+1)/6
• k=3: Σi³ = [n(n+1)/2]²

Performance

• Time Complexity: O(1) for k ∈ {0,1,2,3}, symbolic for k > 3
• Space Complexity: O(1) - constant expression construction

pub fn convergence_test( expr: &Expression, variable: &Symbol, ) -> ConvergenceResult

Check convergence of infinite series

Simplified p-series test: Σ(1/n^p) converges iff p > 1

Performance

• Time Complexity: O(1) - pattern matching only
• Space Complexity: O(1) - returns enum variant