Series Revision Notes for Grade 12 NSC Matric Mathematics

Series

Introduction to series

A series is the sum of the terms in a sequence. When working with sequences, we often need to find the total of several terms added together. This process of adding sequence terms is fundamental in mathematics and appears frequently in NSC examinations.

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Understanding series helps us solve real-world problems involving cumulative totals, such as calculating compound interest, population growth, or determining the total distance travelled over time.

Finite series

A finite series involves adding a specific, limited number of terms from a sequence. We use the symbol Sn to represent the sum of the first n terms of a sequence {T₁; T₂; T₃; ...; Tₙ}.

The general form is: $S_n = T_1 + T_2 + T_3 + ... + T_n$

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Worked Example: Finite Series Calculation

Consider the sequence: 1; 4; 9; 16; 25; 36; 49; ...

To find the sum of the first four terms: $S_4 = 1 + 4 + 9 + 16 = 30$

This demonstrates a finite series because we are only adding four specific terms, not continuing indefinitely.

Infinite series

An infinite series involves adding infinitely many terms from a sequence. We sometimes use the symbol S∞ to represent the sum to infinity (where it exists), or write it using sigma notation as $\sum_{n=1}^{\infty} T_n$ .

The general form is: $S_∞ = T_1 + T_2 + T_3 + ...$

Unlike finite series, infinite series continue without end. While we cannot actually add infinitely many numbers, mathematicians have developed methods to determine what value these sums approach.

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A sum to infinity only exists if the series is convergent. Many infinite series do not have a finite sum (they diverge). For example, a geometric series has a sum to infinity only when $|r| < 1$ .

Sigma notation

Sigma notation provides a compact and efficient way to write series using mathematical symbols. The Greek letter Σ (sigma) represents summation, which makes sense since sigma looks like the letter "S" for "sum".

Components of sigma notation

When we write a sum using sigma notation, several important parts work together:

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General form: $\sum_{i=m}^{n} T_i$

i is the index of the sum (the variable that changes)
m is the lower bound (starting value, shown below the sigma symbol)
n is the upper bound (ending value, shown above the sigma symbol)
T_i represents a term of the sequence
The number of terms in the series equals $(n - m + 1)$

The index i increases from m to n in steps of 1, generating each term to be added. The index can be any variable (commonly $i$ or $n$ ).

Expanded form vs sigma notation

When we write out all individual terms being added, this is called the expanded form. For example:

Expanded form: $2 + 4 + 6 + 8 + 10 = 30$
Sigma notation: $\sum_{n=1}^{5} 2n = 30$

Both expressions represent the same mathematical operation, but sigma notation is more concise.

Rules for sigma notation

Understanding these rules helps you manipulate and solve sigma notation problems efficiently:

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Rule 1: Separating sums

For two sequences $a_i$ and $b_i$ : $\sum_{i=m}^{n} (a_i + b_i) = \sum_{i=m}^{n} a_i + \sum_{i=m}^{n} b_i$

This means you can separate addition inside sigma notation into two separate sums.

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Rule 2: Factoring constants

For any constant $c$ that does not depend on the index $i$ : $\sum_{i=m}^{n} (c \cdot a_i) = c \cdot \sum_{i=m}^{n} a_i$

You can factor out constants from sigma notation, making calculations simpler.

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Rule 3: Proper use of brackets

Always use brackets correctly when substituting values, especially with negative numbers.

Correct: If d = -7, then $T_n = 31 + (n-1)(-7)$
Incorrect: $T_n = 31 + (n-1) - 7$

Worked examples

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Worked Example 1: Expanding Sigma Notation

Question: Expand and find the value of $\sum_{n=1}^{6} 2^n$

Solution:

Step 1: Expand the formula and write the first six terms

$\sum_{n=1}^{6} 2^n = 2^1 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6$

$= 2 + 4 + 8 + 16 + 32 + 64$

This represents a geometric sequence with first term 2 and common ratio 2.

Step 2: Calculate the sum

$S_6 = 2 + 4 + 8 + 16 + 32 + 64 = 126$

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Worked Example 2: Working with Algebraic Expressions

Question: Find the value of $\sum_{n=3}^{7} 2an$

Solution:

Step 1: Expand and write out the five terms

$\sum_{n=3}^{7} 2an = 2a(3) + 2a(4) + 2a(5) + 2a(6) + 2a(7)$

$= 6a + 8a + 10a + 12a + 14a$

Step 2: Simplify by collecting like terms

$\sum_{n=3}^{7} 2an = 50a$

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Worked Example 3: Converting Series to Sigma Notation

Question: Write $31 + 24 + 17 + 10 + 3$ in sigma notation

Solution:

Step 1: Identify the pattern Looking at consecutive terms:

T₁ = 31, T₂ = 24, T₃ = 17, T₄ = 10, T₅ = 3
d = T₂ - T₁ = 24 - 31 = -7
d = T₃ - T₂ = 17 - 24 = -7

This confirms an arithmetic series with common difference d = -7.

Step 2: Find the general term formula

Using $T_n = a + (n-1)d$ :

$T_n = 31 + (n-1)(-7)$

$= 31 - 7(n-1)$

$= 31 - 7n + 7$

$T_n = -7n + 38$

Step 3: Write in sigma notation

$31 + 24 + 17 + 10 + 3 = \sum_{n=1}^{5} (-7n + 38)$

Key formulas (Exam focus)

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Arithmetic series: $S_n = \frac{n}{2}(2a + (n-1)d)$

Geometric series: $S_n = a \frac{1 - r^n}{1 - r}$

Geometric series (sum to infinity): $S_∞ = \frac{a}{1 - r}, \quad |r| < 1$

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Exam tips

Always check your arithmetic: Small calculation errors can lead to incorrect final answers
Use brackets correctly: Especially important when dealing with negative common differences
Identify the pattern first: Determine whether you have an arithmetic or geometric series before proceeding
Show clear working: Write out the expansion before calculating the final sum
Verify your answer: Check that your sigma notation produces the original series when expanded

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Key Points to Remember:

Series are sums of sequence terms - finite series have a set number of terms while infinite series continue forever
Sigma notation uses the symbol Σ to write sums compactly, with the index running from the lower bound to the upper bound
Key components of sigma notation include the index, upper bound, lower bound, and general term
Mathematical rules allow you to separate sums and factor out constants when working with sigma notation
Bracket usage is crucial when substituting negative values to avoid calculation errors
Only convergent series have a sum to infinity

Series (Grade 12 NSC Matric Mathematics): Revision Notes

Series

Introduction to series

Finite series

Infinite series

Sigma notation

Components of sigma notation

Expanded form vs sigma notation

Rules for sigma notation

Worked examples

Key formulas (Exam focus)

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