Summation Calculator

Summation Calculator

Enter the number series or a function with lower and upper limits to compute their sum using the calculator

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Sum (Σ) Sigma Notation

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Summation Calculator

The Summation Calculator helps you find the sum of a number series quickly. It supports both simple numbers and algebraic expressions, with specified lower and upper limits. Step-by-step solutions make it easy to understand the calculations.

What Is Summation?

Definition: Summation is the process of adding a series of numbers (called summands or addends) to obtain a total sum.

A sequence is a series of numbers that defines the addition operation "+".

Summation Symbol: Σ (Greek Letter)

summation

Summation Formula

The general sigma notation is:

\(\sum_{i=1}^{n} x_i = x_1 + x_2 + x_3 + \dots + x_n\)

i = lower bound (starting index)
n = upper bound (ending index)

How to Calculate a Summation

Example #1: Simple Numbers

Find the sum of the first ten composite numbers:

4, 6, 8, 9, 10, 12, 14, 15, 16, 18

Step 1: Add all numbers:

Sum = 4 + 6 + 8 + 9 + 10 + 12 + 14 + 15 + 16 + 18

Answer: Sum = 112

Example #2: Sigma Notation

Evaluate: \(\sum_{i=3}^{7} x_i^3\)

Lower limit = 3, Upper limit = 7
Expand the sigma: \(x_3^3 + x_4^3 + x_5^3 + x_6^3 + x_7^3\)
Insert actual values: \(3^3 + 4^3 + 5^3 + 6^3 + 7^3\)
Compute the sum: \(27 + 64 + 125 + 216 + 343 = 775\)

Types of Summation

1. Simple Series Sum

Example: 2 + 3 + 4 + 5 + 65 + 6 + 6 = 91

Description: Represents an arithmetic sum of numbers without sigma notation.

2. Sigma Notation

Example: \(\sum_{i=0}^{n} f(x_i)\)

Description: Expand from the lower limit (i) to the upper limit (n) to evaluate the total sum.

Important Summation Formulas

Sum Type	Formula
Sum of natural numbers	\(\sum_{x=1}^{m} x = \frac{m(m+1)}{2}\)
Sum of squares	\(\sum_{x=1}^{m} x^2 = \frac{m(m+1)(2m+1)}{6}\)
Sum of cubes	\(\sum_{x=1}^{m} x^3 = \frac{m^2(m+1)^2}{4}\)
Sum of 4th powers	\(\sum_{x=1}^{m} x^4 = \frac{m(m+1)(2m+1)(3m^2 + 3m -1)}{30}\)
Sum of first m even numbers	\(\sum_{x=1}^{m} 2x = m(m+1)\)
Sum of first m odd numbers	\(\sum_{x=1}^{m} (2x-1) = m^2\)
Sum of an arithmetic sequence	\(\sum_{x=1}^{m} a + (x-1)d = \frac{m[2a + (m-1)d]}{2}\)

Double Summation

Compute inner sums first, then outer sums.
Change the order of summation carefully if needed.
Repeat for all terms in the external sum.