1. Introduction to Sigma Notation
Sigma notation is a compact way to write long sums. Instead of writing every term one by one, the Greek letter \( \Sigma \) is used to show that terms are being added according to a pattern or rule.
With sigma notation, a sum that would normally take many symbols can be written in a single expression that clearly shows the starting point, ending point, and the rule for each term.
1.1. Meaning of Σ (Summation)
The symbol \( \Sigma \) (capital sigma) stands for summation, which means "add all the terms" of a certain form.
A general sigma expression looks like:
\( \displaystyle \sum_{k = 1}^{n} a_k \)
This means: "Add all the terms \( a_k \) starting from \( k = 1 \) up to \( k = n \)."
1.2. How to Read a Sigma Expression
A sigma expression has three main parts:
- The symbol \( \Sigma \) tells you to add.
- The index (like \( k \)) with a starting value below, such as \( k = 1 \).
- The ending value above, such as \( n \).
For example:
\( \displaystyle \sum_{k = 1}^{4} k = 1 + 2 + 3 + 4 \)
So, sigma notation is just a short way of writing a long sum.
2. Sum of First n Natural Numbers
One of the most famous special series is the sum of the first \( n \) natural numbers. Natural numbers here mean the counting numbers starting from 1: \( 1, 2, 3, \ldots, n \).
2.1. Formula for \(\sum_{k=1}^{n} k\)
The sum of the first \( n \) natural numbers is given by:
\( \displaystyle \sum_{k=1}^{n} k = 1 + 2 + 3 + \cdots + n = \dfrac{n(n+1)}{2} \)
This formula shows that the total grows roughly like \( n^2 \) when \( n \) becomes large.
2.2. Example
Find \( 1 + 2 + 3 + \cdots + 20 \).
Using the formula:
\( \displaystyle \sum_{k=1}^{20} k = \dfrac{20 \cdot 21}{2} = 210 \)
So the sum of the first 20 natural numbers is 210.
3. Sum of Squares of First n Natural Numbers
The next special series deals with the squares of the natural numbers, like \( 1^2, 2^2, 3^2, \ldots, n^2 \). This sum appears often in problems related to growth, area, or variations.
3.1. Formula for \(\sum_{k=1}^{n} k^2\)
The sum of the squares of the first \( n \) natural numbers is:
\( \displaystyle \sum_{k=1}^{n} k^2 = 1^2 + 2^2 + 3^2 + \cdots + n^2 = \dfrac{n(n+1)(2n+1)}{6} \)
This formula grows like \( n^3 \) when \( n \) becomes large.
3.2. Example
Find \( 1^2 + 2^2 + 3^2 + 4^2 + 5^2 \).
Here, \( n = 5 \).
\( \displaystyle \sum_{k=1}^{5} k^2 = \dfrac{5 \cdot 6 \cdot 11}{6} = 55 \)
This matches the direct sum: \( 1 + 4 + 9 + 16 + 25 = 55 \).
4. Sum of Cubes of First n Natural Numbers
The third special series uses cubes of natural numbers: \( 1^3, 2^3, 3^3, \ldots, n^3 \). The formula for this sum has a very beautiful pattern.
4.1. Formula for \(\sum_{k=1}^{n} k^3\)
The sum of the cubes of the first \( n \) natural numbers is:
\( \displaystyle \sum_{k=1}^{n} k^3 = 1^3 + 2^3 + 3^3 + \cdots + n^3 = \left( \dfrac{n(n+1)}{2} \right)^2 \)
So, the sum of cubes is actually the square of the sum of the first \( n \) natural numbers.
4.2. Example
Find \( 1^3 + 2^3 + 3^3 + 4^3 \).
First, find \( \displaystyle \dfrac{n(n+1)}{2} \) for \( n = 4 \):
\( \dfrac{4 \cdot 5}{2} = 10 \)
Now square it:
\( 10^2 = 100 \)
So \( \displaystyle \sum_{k=1}^{4} k^3 = 100 \).
5. Patterns in Special Series
These special series are not isolated formulas; they are connected by simple and elegant patterns. Noticing these patterns makes the formulas much easier to remember and understand.
5.1. Square of the Sum Pattern
A key pattern is:
\( \displaystyle \sum_{k=1}^{n} k^3 = \left( \sum_{k=1}^{n} k \right)^2 \)
In words: the sum of the cubes of the first \( n \) natural numbers is the square of the sum of the first \( n \) natural numbers.
This simple relationship makes the cube formula much easier to recall.
5.2. Visual Understanding
These sums can be imagined using dot or block patterns:
- \( \sum k \): stacking rows of dots to form a triangular shape.
- \( \sum k^2 \): arranging small squares into larger square or rectangular patterns.
- \( \sum k^3 \): building cube-like stacks whose total number of unit cubes matches a perfect square.
Thinking visually helps connect algebraic formulas with geometric shapes and makes the patterns feel more natural.