Introduction to summation explained with examples

Everyone is familiar with the summation term. Since the start of human beings, everyone has been conscious of how they add the terms conveniently. By removing the need to jot down a list of series terms, summation notation saves time. In this post, we’ll go through some new strategies for adding large and complicated functions that are tough by using summation.

Simple Summation

The summation function is used to add up the values of the provided data. We add them and find the solution to the data. Simple summation is applied for small data values and it is quite difficult to find the summation of large data set in this method.

Example:

Find the sum of 2, 3, 6, 7, and 8.

2 + 3 + 6 + 7 + 8 = 26

What is Sigma?

Sigma notation is used to find the summation of the given series or any function. This notation is very convenient to write the large data in a small notation. It provides detailed information about what we should add up. Summation notation saves time by eliminating the need to write down a list of series terms.

The result of the summation is a single numerical number. The sigma notation gives us specific information about the sequence which we add up. The notation of sigma is `\sum_{i=1}^n X_i`

1. i is called the index of the summation.
2. 1 is the starting value of summation or the lower limit of summation.
3. n is called the stopping point or upper limit of the summation.
4. `\sum` is a sign of summation.
5. `x_i` is called the element

Some of the Properties of the summation

`\sum_{n=i}^t C.f(x)`	`C.\sum_{n=i}^t f(x)`
`\sum_{n=i}^t [f(x)+g(x)]`	`\sum_{n=i}^t f(x)+\sum_{n=i}^t g(x)`
`\sum_{n=i}^t [f(x)-g(x)]`	`\sum_{n=i}^t f(x)-\sum_{n=i}^t g(x)`
`\sum_{n=i+p}^{t+p} f(x-p)`	`\sum_{n=i+p}^{t+p} f(x)`
`\sum_{n=i}^t\sum_{k=i}^l b_{n.k}`	`\sum_{k=i}^l\sum_{n=i}^t b_{n.k}`
`\prod_{n=i}^t ln f(x)`	`ln \prod_{n=i}^t f(x)`

Constant terms and functions summation

Let us we have a constant number b and we want to add n term of the constant b. it is given as the n times of b.

`\sum_{i=1}^n C` = C + C + C + C +…n times

`\sum_{i=1}^n C` = n x C

e.g. if we have added value 512 times.

5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 60

By using the rule it is convent to solve as

C x n

Where c is a constant term and n is the number of times it appears.

5 x 12 = 60

Examples of summation

Here are a few examples of sigma notation problems handled utilizing Summation's principles and formulae.

Example 1:

Find the sum of terms of 'x' if 'x' varies from 1 to 10.

Solution:

Step 1: We have to write in sigma notation

`\sum_{x=1}^10 X`

Step 2: Apply the summation symbol now.

`\sum_{x=1}^10 X` = 1+2+3+4+5+6+7+8+9+10

Step 3: Add all the resultant terms

`\sum_{x=1}^10 X` = 55

So the sum of the first 10 terms is 55.

With the help of a summation calculator, we may quickly and easily determine the answer to any series. Step by step solution is provided which develops the logical view for the reader.

Summation Calculator

• First select the summation options.
• If we choice sigma notation then we have a multiply value in the view bar.
• Enter the variable x's initial value.
• Put the equation together.
• The ultimate value of n
• Select the variable.
• Press the enter key or calculation button.

We have the solution of the given data value we have entered.

Example 2:

Find the sum of series x² + 2 x [cos(x) + `\sqrt{x}`] by using sigma method where x start from 1 and ends on 9.

Solution:

Step 1: Change into sigma notation

`\sum_{x=1}^9 x^2+2\times[cos(x)+\sqrt{x}]`

Step 2: Apply the summation

Value of x	Equation	Evaluate
1.	`1^2+2\times[cos(1)+\sqrt{1}]`	4.0806
2.	`2^2+2\times[cos(2)+\sqrt{2}]`	5.9961
3.	`3^2+2\times[cos(3)+\sqrt{3}]`	10.4841
4.	`4^2+2\times[cos(4)+\sqrt{4}]`	18.6927
5.	`5^2+2\times[cos(5)+\sqrt{5}]`	30.03946
6.	`6^2+2\times[cos(6)+\sqrt{6}]`	42.8193
7.	`7^2+2\times[cos(7)+\sqrt{7}]`	55.9733
8.	`8^2+2\times[cos(8)+\sqrt{8}]`	69.3659
9.	`9^2+2\times[cos(9)+\sqrt{9}]`	85.1778
`\sum_{x=1}^9`	Total Sum	322.4553

The total sum of the series x² + 2 x [cos(x) + `\sqrt{x}`] =322.4553

Example 3:

Find the sum of `cos(\frac{\pi x}{10})` where the value of x starts from 1 and ends at 10.

Solution:

Step 1: Change into sigma notation

`\sum_{x=1}^10 cos(\frac{\pi x}{10})`

Step 2: Draw a table by varying the value of x from 1 to 10 with the evaluation of the function.

Value of x	Equation	Evaluate
1.	`cos(\frac{\pi.1}{10})`	0.9511
2.	`cos(\frac{\pi.2}{10})`	0.8090
3.	`cos(\frac{\pi.3}{10})`	0.5878
4.	`cos(\frac{\pi.4}{10})`	0.3090
5.	`cos(\frac{\pi.5}{10})`	6.1232x10-17
6.	`cos(\frac{\pi.6}{10})`	-0.3090
7.	`cos(\frac{\pi.7}{10})`	-0.5878
8.	`cos(\frac{\pi.8}{10})`	-0.8090
9.	`cos(\frac{\pi.9}{10})`	-0.9511
10.	`cos(\frac{\pi.10}{10})`	-1
`\sum_{x=1}^10`	Total Sum	-0.9999

Hence `\sum_{x=1}^10 cos (\frac{\pi x}{10})` = -0.9999

Summary

In this article, we have learnt about the summation, sigma notation, properties, and detailed examples with step-by-step solutions. You are able now to add any series, function, and complex problems in an easier way.