Series
Complete Notes & Formulae for Eleventh Grade (Algebra 2)
1. Identify Arithmetic and Geometric Series
What is a Series?
A series is the sum of the terms of a sequence
Sequence: 2, 4, 6, 8, 10
Series: 2 + 4 + 6 + 8 + 10 = 30
Arithmetic Series:
Definition:
Sum of terms in an arithmetic sequence (constant difference between terms)
General Form:
\[ a + (a+d) + (a+2d) + (a+3d) + \cdots \]
where \( a \) = first term, \( d \) = common difference
How to Identify:
• Check if there's a constant difference between consecutive terms
• \( d = a_{n+1} - a_n \) is the same for all terms
Example:
3 + 7 + 11 + 15 + 19 (common difference = 4)
Geometric Series:
Definition:
Sum of terms in a geometric sequence (constant ratio between terms)
General Form:
\[ a + ar + ar^2 + ar^3 + \cdots \]
where \( a \) = first term, \( r \) = common ratio
How to Identify:
• Check if there's a constant ratio between consecutive terms
• \( r = \frac{a_{n+1}}{a_n} \) is the same for all terms
Example:
2 + 6 + 18 + 54 + 162 (common ratio = 3)
2. Introduction to Sigma Notation (Σ)
Sigma Notation:
A compact way to represent the sum of a series using the Greek letter Σ (sigma)
\[ \sum_{i=1}^{n} a_i = a_1 + a_2 + a_3 + \cdots + a_n \]
Parts of Sigma Notation:
• \( i \) = index of summation (variable)
• \( 1 \) = lower limit (starting value)
• \( n \) = upper limit (ending value)
• \( a_i \) = general term (formula for each term)
Examples:
Example 1: \( \sum_{i=1}^{5} i \)
= 1 + 2 + 3 + 4 + 5 = 15
Example 2: \( \sum_{k=1}^{4} 2k \)
= 2(1) + 2(2) + 2(3) + 2(4) = 2 + 4 + 6 + 8 = 20
Example 3: \( \sum_{n=0}^{3} 3^n \)
= 3^0 + 3^1 + 3^2 + 3^3 = 1 + 3 + 9 + 27 = 40
3. Find the Sum of an Arithmetic Series
Formula:
\[ S_n = \frac{n}{2}(a_1 + a_n) \]
or equivalently
\[ S_n = \frac{n}{2}[2a + (n-1)d] \]
where:
• \( S_n \) = sum of first n terms
• \( n \) = number of terms
• \( a \) or \( a_1 \) = first term
• \( a_n \) = last term
• \( d \) = common difference
Example:
Find the sum: 5 + 8 + 11 + 14 + ... (10 terms)
Given: \( a_1 = 5, d = 3, n = 10 \)
Step 1: Find \( a_{10} = a_1 + (n-1)d = 5 + 9(3) = 32 \)
Step 2: \( S_{10} = \frac{10}{2}(5 + 32) = 5(37) = 185 \)
Answer: 185
4. Find the Sum of a Finite Geometric Series
Formula:
\[ S_n = a \cdot \frac{1 - r^n}{1 - r} \quad \text{when } r \neq 1 \]
or equivalently
\[ S_n = a \cdot \frac{r^n - 1}{r - 1} \quad \text{when } r \neq 1 \]
where:
• \( S_n \) = sum of first n terms
• \( a \) = first term
• \( r \) = common ratio
• \( n \) = number of terms
⚠️ If \( r = 1 \), then \( S_n = na \)
Example:
Find the sum: 3 + 6 + 12 + 24 + 48
Given: \( a = 3, r = 2, n = 5 \)
\( S_5 = 3 \cdot \frac{1 - 2^5}{1 - 2} = 3 \cdot \frac{1 - 32}{-1} = 3 \cdot \frac{-31}{-1} = 3 \cdot 31 = 93 \)
Answer: 93
5. Introduction to Partial Sums
What are Partial Sums?
A partial sum is the sum of the first n terms of a series
Notation:
\[ S_n = \sum_{i=1}^{n} a_i \]
Sequence of Partial Sums:
\( S_1 = a_1 \)
\( S_2 = a_1 + a_2 \)
\( S_3 = a_1 + a_2 + a_3 \)
\( S_n = a_1 + a_2 + a_3 + \cdots + a_n \)
Partial Sums of Arithmetic Series:
\[ S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}[2a + (n-1)d] \]
Partial Sums of Geometric Series:
\[ S_n = a \cdot \frac{1 - r^n}{1 - r} \quad (r \neq 1) \]
6. Convergent and Divergent Geometric Series
Convergence Test:
An infinite geometric series converges or diverges based on the common ratio \( r \)
CONVERGES (has a finite sum):
\[ |r| < 1 \quad \text{or} \quad -1 < r < 1 \]
The terms get smaller and approach zero
DIVERGES (no finite sum):
\[ |r| \geq 1 \]
The terms don't approach zero; series grows without bound
Examples:
Series: \( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots \)
\( r = \frac{1}{2} \), and \( |\frac{1}{2}| < 1 \)
CONVERGES ✓
Series: \( 1 + 2 + 4 + 8 + 16 + \cdots \)
\( r = 2 \), and \( |2| > 1 \)
DIVERGES ✗
Series: \( 3 - \frac{3}{4} + \frac{3}{16} - \frac{3}{64} + \cdots \)
\( r = -\frac{1}{4} \), and \( |-\frac{1}{4}| < 1 \)
CONVERGES ✓
7. Find the Value of an Infinite Geometric Series
Formula for Infinite Geometric Series:
When \( |r| < 1 \), the infinite series converges to:
\[ S = \frac{a}{1 - r} \]
where:
• \( S \) = sum of the infinite series
• \( a \) = first term
• \( r \) = common ratio (\( |r| < 1 \))
Examples:
Example 1: Find sum of \( 6 + 3 + \frac{3}{2} + \frac{3}{4} + \cdots \)
First term: \( a = 6 \)
Common ratio: \( r = \frac{3}{6} = \frac{1}{2} \)
Check: \( |\frac{1}{2}| < 1 \) ✓ (converges)
\( S = \frac{6}{1 - \frac{1}{2}} = \frac{6}{\frac{1}{2}} = 12 \)
Answer: 12
Example 2: Find sum of \( \sum_{n=0}^{\infty} 5\left(\frac{2}{3}\right)^n \)
First term: \( a = 5 \)
Common ratio: \( r = \frac{2}{3} \)
Check: \( |\frac{2}{3}| < 1 \) ✓ (converges)
\( S = \frac{5}{1 - \frac{2}{3}} = \frac{5}{\frac{1}{3}} = 15 \)
Answer: 15
8. Write a Repeating Decimal as a Fraction
Method: Use Infinite Geometric Series
A repeating decimal can be expressed as an infinite geometric series
Steps:
1. Identify the repeating part
2. Write as a geometric series
3. Find \( a \) and \( r \)
4. Use formula \( S = \frac{a}{1-r} \)
5. Simplify to lowest terms
Examples:
Example 1: \( 0.\overline{3} = 0.333... \)
Write as: \( 0.3 + 0.03 + 0.003 + \cdots \)
This is: \( \frac{3}{10} + \frac{3}{100} + \frac{3}{1000} + \cdots \)
First term: \( a = \frac{3}{10} \)
Common ratio: \( r = \frac{1}{10} \)
\( S = \frac{\frac{3}{10}}{1 - \frac{1}{10}} = \frac{\frac{3}{10}}{\frac{9}{10}} = \frac{3}{10} \times \frac{10}{9} = \frac{3}{9} = \frac{1}{3} \)
Answer: \( \frac{1}{3} \)
Example 2: \( 0.\overline{27} = 0.272727... \)
Write as: \( 0.27 + 0.0027 + 0.000027 + \cdots \)
This is: \( \frac{27}{100} + \frac{27}{10000} + \frac{27}{1000000} + \cdots \)
First term: \( a = \frac{27}{100} \)
Common ratio: \( r = \frac{1}{100} \)
\( S = \frac{\frac{27}{100}}{1 - \frac{1}{100}} = \frac{\frac{27}{100}}{\frac{99}{100}} = \frac{27}{99} = \frac{3}{11} \)
Answer: \( \frac{3}{11} \)
9. Quick Reference Summary
Key Formulas:
Arithmetic Series Sum:
\[ S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}[2a + (n-1)d] \]
Finite Geometric Series Sum:
\[ S_n = a \cdot \frac{1 - r^n}{1 - r} \quad (r \neq 1) \]
Infinite Geometric Series Sum:
\[ S = \frac{a}{1 - r} \quad (|r| < 1) \]
Sigma Notation:
\[ \sum_{i=1}^{n} a_i = a_1 + a_2 + \cdots + a_n \]
Convergence Test:
Converges if \( |r| < 1 \)
Diverges if \( |r| \geq 1 \)
| Type | Pattern | Key Formula |
|---|---|---|
| Arithmetic | Constant difference (d) | \( S_n = \frac{n}{2}(a_1 + a_n) \) |
| Geometric (Finite) | Constant ratio (r) | \( S_n = a \cdot \frac{1-r^n}{1-r} \) |
| Geometric (Infinite) | |r| < 1 | \( S = \frac{a}{1-r} \) |
📚 Study Tips
✓ Identify the type of series before choosing a formula
✓ For geometric series, always check if |r| < 1 for convergence
✓ Remember: arithmetic uses addition, geometric uses multiplication
✓ Sigma notation is just a compact way to write a sum
✓ Repeating decimals are infinite geometric series!