4.5.2 Sigma Notation

Sigma Notation

\sum_{n=1}^{4} 2n + 1

Sigma notation is a compact way of expressing the sum of a series of terms. It uses the Greek letter $\Sigma$ (capital sigma), which stands for "sum."

These are called limits.

This means to evaluate $2n + 1$ for $n = 1$ to $n = 4, n \in \mathbb{N}$ .

2(1) + 1 + 2(2) + 1 + 2(3) + 1 + 2(4) + 1 = :highlight[24]

e.g. Find $\sum_{n=1}^{4} 2$

This refers to the variable that we are incrementing. If the variable does not appear in the expression, we still evaluate the expression.

2 + 2 + 2 + 2 = :highlight[8]

With each term corresponding to $n = 1, 2, 3, 4$ .

General Form:

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The sigma notation is written as: $\sum_{k=m}^{n} a_k$

This represents the sum of the terms $\ a_k$ from $\ k = m \ to \ k = n .$ Here's what each part means:

$\ \Sigma$ denotes summation.
$\ k$ is the index of summation, which starts at the lower limit $\ m$ and increases to the upper limit $\ n .$
$\ a_k$ is the general term of the sequence you're summing.

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Example 1:

Sum the first $5$ natural numbers:

$\sum_{k=1}^{5} k = 1 + 2 + 3 + 4 + 5 = :highlight[15]$

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Example 2:

Sum the squares of the first 4 natural numbers:

$\sum_{k=1}^{4} k^2 = 1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = :highlight[30]$

Step-by-Step Breakdown:

Identify the General Term $\ a_k :$ This is the expression that changes as $\ k$ changes. For example, in $\ \sum_{k=1}^{4} k^2$ , the general term $\ a_k = k^2 .$
Determine the Limits of Summation: The lower limit $m$ is where $\ k$ starts, and the upper limit $\ n$ is where $\ k$ ends. In the example $\ \sum_{k=1}^{4} k^2 \, \ k$ starts at $1$ and ends at $4$ .
Expand and Calculate: Substitute the values of $\ k \ from \ m \ to \ n$ into the general term and add them together.

Properties of Sigma Notation:

Linearity: $\sum_{k=m}^{n} (a_k + b_k) = \sum_{k=m}^{n} a_k + \sum_{k=m}^{n} b_k$ $\sum_{k=m}^{n} c \cdot a_k = c \cdot \sum_{k=m}^{n} a_k$ where $\ c$ is a constant.
Shifting the Index: If you shift the index by a constant $\ c$ , the sum changes accordingly: $\sum_{k=m}^{n} a_k = \sum_{j=m+c}^{n+c} a_{j-c}$ where $\ j = k + c .$
Splitting the Sum: $\sum_{k=m}^{n} a_k = \sum_{k=m}^{p} a_k + \sum_{k=p+1}^{n} a_k$ for any integer $\ p$ between $\ m \ and \ n .$

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Example Exam Question:

Question: Evaluate the sum:

$\sum_{k=1}^{6} (2k + 1)$

[3 marks]

Solution:

Expand the Summation: $\sum_{k=1}^{6} (2k + 1) = (2(1) + 1) + (2(2) + 1) + (2(3) + 1) + (2(4) + 1) + (2(5) + 1) + (2(6) + 1)$ $= 3 + 5 + 7 + 9 + 11 + 13$
Calculate the Sum: $3 + 5 + 7 + 9 + 11 + 13 = :highlight[48]$ Final Answer: $\sum_{k=1}^{6} (2k + 1) = :highlight[48]$

Sigma Notation Simplified Revision Notes for A-Level AQA Maths Pure

4.5.2 Sigma Notation

Sigma Notation

These are called limits.

General Form:

Example 1:

Example 2:

Step-by-Step Breakdown:

Properties of Sigma Notation:

Example Exam Question:

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