Sum of Series: Mastering Sigma Notation and Convergence

The sum of a series — whether finite or infinite — is one of the most important concepts in mathematics. From computing areas under curves to analyzing algorithms, series provide the tools to handle accumulated quantities.

Sigma Notation

Sigma notation provides a compact way to write sums. The index k runs from the lower limit to the upper limit.

Common Series Formulas

• Arithmetic: Sₙ = n/2 × (2a₁ + (n-1)d)
• Geometric: Sₙ = a₁ × (1 - rⁿ)/(1 - r)
• Squares: Σ k² = n(n+1)(2n+1)/6
• Cubes: Σ k³ = [n(n+1)/2]²

Infinite Series and Convergence

An infinite series converges if its partial sums approach a finite limit. The geometric series Σ rᵏ converges to 1/(1-r) when |r| < 1.

Worked Example

Sum of the first 50 squares:

1. Use Σ k² = n(n+1)(2n+1)/6
2. S₅₀ = 50 × 51 × 101 / 6
3. = 257550 / 6 = 42925

Convergence Tests

• Ratio test: Check |aₙ₊₁/aₙ|
• Comparison test: Compare with known convergent series
• Integral test: Relate series to improper integral
• Alternating series test: For series with alternating signs