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# Summation Calculator

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

Calculate Sum Method:

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.

## What Is Summation?

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)

## Summation Formula:

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

## How to Calculate a Summation?

### Example # 01:

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:

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

Sum = 112

### Example # 02:

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

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

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

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

## Types of Summation:

Summation is of two types that include:

### Simple Series Sum:

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

#### Description:

Simple summation represents a simple arithmetic sum of numbers.

### Sigma Notation:

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

Description:

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).

## Important Summation Equations:

 Sigma (Summation) for Formula Sum of natural numbers mΣx=1 x = [m(m + 1)]/2 Sum of squares of natural numbers mΣx=1 x2 = [m(m + 1)(2m + 1)]/6 Sum of cubes of natural numbers mΣx=1 x3 = [m2(m + 1)2]/4 Sum of 4th power of natural numbers mΣx=1 x4 = [m(m + 1)(2m + 1)(3m2 + 3m – 1)]/30 Sum of 1st m even numbers mΣx=1 2x = m(m + 1) Sum of 1st m odd numbers mΣx=1 (2x + 1) = m2 Sum of an arithmetic sequence mΣx=1 a + (x – 1)d = m[2a + (m – 1)d]/2

## How Do I Do Double Summation?

• 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