Sum of Series Calculator
Calculate finite and infinite sums of arithmetic and geometric series
Calculate Series Sum
Quick Examples
Calculation Results
Series Preview:
Step-by-Step Calculation:
Arithmetic Series: a = 1, d = 1, n = 10
Formula: Sₙ = n/2 × [2a + (n-1)d]
S10 = 10/2 × [2×1 + (10-1)×1]
S10 = 5 × [2 + 9]
S10 = 5 × 11 = 55
Example: First 10 Odd Numbers
Problem
Find the sum: 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19
This is an arithmetic series with first term a = 1 and common difference d = 2
Solution
Given: a = 1, d = 2, n = 10
Formula: Sₙ = n/2 × [2a + (n-1)d]
Calculation: S₁₀ = 10/2 × [2×1 + (10-1)×2]
Result: S₁₀ = 5 × [2 + 18] = 5 × 20 = 100
Key Formulas
Arithmetic Series
Sₙ = n/2 × [2a + (n-1)d]
Or: Sₙ = n/2 × (first + last)
Geometric Series (Finite)
Sₙ = a × (1-rⁿ)/(1-r)
For r ≠ 1
Geometric Series (Infinite)
S = a/(1-r)
Only if |r| < 1
Convergence Guide
|r| < 1
Series converges to finite sum
|r| > 1
Series diverges to infinity
|r| = 1
Series is periodic
Arithmetic
No infinite sum (except d=0)
Understanding Series and Their Sums
What is a Series?
A series is the sum of the terms in a sequence. We can calculate finite sums (partial sums) or, under certain conditions, infinite sums where the series converges to a finite value.
Types of Series
- •Arithmetic: Constant difference between terms
- •Geometric: Constant ratio between terms
- •Harmonic: Reciprocals of arithmetic sequence
- •Power: Terms involve powers of position
Applications
Finance
Loan payments, compound interest, annuities
Physics
Wave functions, oscillations, decay processes
Engineering
Signal processing, control systems, approximations
Computer Science
Algorithm analysis, complexity theory, fractals