Summation Calculator
Calculate the summation (Σ) of a series from i = start to n. Choose from preset expressions including i, i², i³, 2i, 2i+1, and 1/i. This calculator evaluates sigma notation for arithmetic, quadratic, cubic, and harmonic series.
How to Calculate a Summation
- Set the starting index value (typically 1).
- Set the ending index value (n).
- Choose the expression to evaluate at each step.
- The calculator evaluates f(i) for each integer from start to end and sums all values.
- For known series, you can verify against closed-form formulas.
Formula
Σ from i=start to n of f(i) = f(start) + f(start+1) + ... + f(n)
Closed-form formulas (starting from i=1):
Σi = n(n+1)/2
Σi² = n(n+1)(2n+1)/6
Σi³ = [n(n+1)/2]²
Σ2i = n(n+1)
Σ(2i+1)= n² + 2n (sum of first n odd numbers starting from 3)
Σ(1/i) = Hₙ (harmonic number, no simple closed form)Example
Calculate Σ from i=1 to 10 of i (sum of first 10 natural numbers):
Σ i from i=1 to 10
= 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10
= 55
Verification with closed form:
n(n+1)/2 = 10(11)/2 = 110/2 = 55 ✓Common Summation Formulas Reference Table
| Series | Closed Form | n=5 | n=10 | n=100 |
|---|---|---|---|---|
| Σi | n(n+1)/2 | 15 | 55 | 5,050 |
| Σi² | n(n+1)(2n+1)/6 | 55 | 385 | 338,350 |
| Σi³ | [n(n+1)/2]² | 225 | 3,025 | 25,502,500 |
| Σ2i | n(n+1) | 30 | 110 | 10,100 |
| Σ(2i+1) | n² + 2n | 35 | 120 | 10,200 |
| Σ(1/i) | Hₙ (harmonic) | 2.2833 | 2.9290 | 5.1874 |
All sums computed from i=1 to n
Frequently Asked Questions
What is sigma (Σ) notation?
Sigma notation (Σ) is a compact way to write the sum of a series. The expression Σ from i=1 to n of f(i) means: evaluate f(i) for each integer i from 1 to n, then add all the results together.
What is the formula for an arithmetic series?
The sum of an arithmetic series with first term a, last term l, and n terms is: S = n(a + l)/2. For the natural numbers 1 to n, this simplifies to n(n+1)/2. This was famously discovered by young Gauss.
What is a geometric series?
A geometric series has a constant ratio between consecutive terms: Σ ar^i from i=0 to n = a(1-r^(n+1))/(1-r) for r≠1. If |r| < 1, the infinite sum converges to a/(1-r).
When is summation used in real life?
Summation is used in statistics (computing means, variances), finance (present value of cash flows), physics (discrete approximations of integrals), computer science (algorithm analysis), and engineering (signal processing).
What is the harmonic series?
The harmonic series is Σ(1/i) from i=1 to infinity. It diverges (grows without bound), but very slowly. The partial sum Hₙ ≈ ln(n) + 0.5772 (Euler-Mascheroni constant). It appears in probability, number theory, and algorithm analysis.