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**Table of Content**

Find the sum of a number series with the summation calculator. The tool also supports the summation of algebraic expressions with lower and upper ranges entered.

With that, get step-wise solution also helps you to understand the calculations more deeply.

In Mathematics:

**“Summation is the addition process of any numbers called the summands or addends that result in the sum or total”**

The sequence is the series that defines the mathematical operation **“+”**.

Summation **Symbol:** **Σ** (A Greek Letter)

The basic sigma equation is as follows:

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

Where:

**i**= Lower bound**n**= Upper bound

Suppose we have the first ten composite numbers listed as:

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

Their sum is calculated as follows:

**Step # 01: Write all numbers with an addition sign in between them**

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

**Answer:**

Sum = 112

If you have a given expression in the sigma notation below:

\(\sum_{n=3}^7 x_{i}^3\)

You may evaluate summation by expanding the sigma notation, which can be done as follows:

**Step # 01: Write down the lower and upper limits**

- Lower limit = 3
- Upper limit = 7

**Step # 02: Now write the original function in the summation notation**

\(\sum_{n=3}^7 x_{i}^3 = x_{3}^3 + x_{4}^3 + x_{5}^3 + x_{6}^3 + x_{7}^3\)

**Step # 03: Enter the actual values**

\(\sum_{n=3}^7 x_{i}^3 = 3^3 + 4^3 + 5^3 + 6^3 + 7^3\)

**Step # 04: Solve to the most simple sigma notation**

\(\sum_{n=3}^7 x_{i}^3 = 3^3 + 4^3 + 5^3 + 6^3 + 7^3\)

\(\sum_{n=3}^7 x_{i}^3 = 27 + 64 + 125 + 216 + 343\)

\(\sum_{n=3}^7 x_{i}^3 = 775\)

Summation is of two types that include:

**2+3+4+5+65+6+6=91**

Simple summation represents a simple arithmetic sum of numbers.

\(\sum_{i=0}^{n} [f\left(x\right)]\)

This formula is expanded to evaluate the final sum. We have to start from the Index (Lower Limit) and terminate at the Endpoint (Upper Limit).

Sigma (Summation) for |
Formula |

Sum of natural numbers | ^{m}Σ_{x=1} x = [m(m + 1)]/2 |

Sum of squares of natural numbers | ^{m}Σ_{x=1} x^{2} = [m(m + 1)(2m + 1)]/6 |

Sum of cubes of natural numbers | ^{m}Σ_{x=1} x^{3} = [m^{2}(m + 1)^{2}]/4 |

Sum of 4^{th} power of natural numbers |
^{m}Σ_{x=1} x^{4} = [m(m + 1)(2m + 1)(3m^{2} + 3m – 1)]/30 |

Sum of 1^{st} m even numbers |
^{m}Σ_{x=1} 2x = m(m + 1) |

Sum of 1^{st} m odd numbers |
^{m}Σ_{x=1} (2x + 1) = m^{2} |

Sum of an arithmetic sequence | ^{m}Σ_{x=1} a + (x – 1)d = m[2a + (m – 1)d]/2 |

- First, change the order of expression for double sums
- Now, the external-sum index is holding and increases the internal index
- After using the internal sum index, increase the external sum index
- Repeat the previous steps for the entire external sum index