Sequence and Series Summation Calculator
Sequence Terms:
Sum:
Solution Steps:
Understanding Sequences and Series
Types of Sequences
Arithmetic Sequence
aₙ = a₁ + (n-1)d
Sₙ = n/2[2a₁ + (n-1)d]
Example: 2, 5, 8, 11, 14, ...
Common difference (d) = 3
Geometric Sequence
aₙ = a₁rⁿ⁻¹
Sₙ = a₁(1-rⁿ)/(1-r)
Example: 2, 6, 18, 54, ...
Common ratio (r) = 3
Special Series
Harmonic Series
Σ(1/n) = 1 + 1/2 + 1/3 + ...
Divergent series with important applications
Power Series
Σ(xⁿ) = 1 + x + x² + x³ + ...
Convergent for |x| < 1
Convergence Tests
Ratio Test
lim|aₙ₊₁/aₙ| < 1
Root Test
lim|aₙ|^(1/n) < 1
Comparison Test
Compare with known series
Applications
Finance
- Compound Interest
- Loan Amortization
- Investment Growth
Physics
- Wave Functions
- Quantum Mechanics
- Statistical Mechanics
Computer Science
- Algorithm Analysis
- Data Structures
- Recursive Functions