Series
Complete Notes & Formulae for Twelfth Grade (Precalculus)
1. What is a Series?
Definition:
A series is the sum of the terms of a sequence
If sequence is: \( a_1, a_2, a_3, \ldots, a_n \)
Then series is: \( a_1 + a_2 + a_3 + \cdots + a_n \)
• Sequence: List of numbers
• Series: Sum of those numbers
2. Arithmetic Series
Definition:
An arithmetic series is the sum of terms in an arithmetic sequence (constant difference between consecutive terms)
Example: 2 + 5 + 8 + 11 + 14 (common difference d = 3)
Sum Formula:
Formula 1: When last term is known
\[ S_n = \frac{n}{2}(a_1 + a_n) \]
Formula 2: When last term is not known
\[ S_n = \frac{n}{2}[2a_1 + (n-1)d] \]
where:
• \( S_n \) = sum of first n terms
• \( a_1 \) = first term
• \( a_n \) = last (nth) term
• \( n \) = number of terms
• \( d \) = common difference
Example:
Find sum: 3 + 7 + 11 + 15 + ... (10 terms)
Given: \( a_1 = 3 \), \( d = 4 \), \( n = 10 \)
Use formula: \( S_n = \frac{n}{2}[2a_1 + (n-1)d] \)
\( S_{10} = \frac{10}{2}[2(3) + (10-1)(4)] \)
\( S_{10} = 5[6 + 36] = 5(42) \)
Sum = 210
3. Geometric Series
Definition:
A geometric series is the sum of terms in a geometric sequence (constant ratio between consecutive terms)
Example: 2 + 6 + 18 + 54 + 162 (common ratio r = 3)
Finite Geometric Series Sum:
\[ S_n = \frac{a_1(1 - r^n)}{1 - r} \quad \text{or} \quad S_n = \frac{a_1(r^n - 1)}{r - 1} \]
where:
• \( S_n \) = sum of first n terms
• \( a_1 \) = first term
• \( r \) = common ratio (r ≠ 1)
• \( n \) = number of terms
Example:
Find sum: 5 + 10 + 20 + 40 + 80
Given: \( a_1 = 5 \), \( r = 2 \), \( n = 5 \)
Use formula: \( S_n = \frac{a_1(r^n - 1)}{r - 1} \)
\( S_5 = \frac{5(2^5 - 1)}{2 - 1} = \frac{5(32 - 1)}{1} = \frac{5(31)}{1} \)
Sum = 155
4. Sigma Notation (Σ)
Definition:
Sigma notation is a compact way to represent a series using the Greek letter Σ (sigma)
\[ \sum_{i=1}^{n} a_i = a_1 + a_2 + a_3 + \cdots + a_n \]
• \( \sum \) = summation symbol (sigma)
• \( i \) = index of summation
• \( i = 1 \) = lower bound (starting value)
• \( n \) = upper bound (ending value)
• \( a_i \) = general term
Examples:
Evaluate: \( \sum_{i=1}^{5} 2i \)
Expand: \( 2(1) + 2(2) + 2(3) + 2(4) + 2(5) \)
= \( 2 + 4 + 6 + 8 + 10 \)
= 30
Write in sigma notation: 1 + 4 + 9 + 16 + 25
Pattern: These are perfect squares \( 1^2, 2^2, 3^2, 4^2, 5^2 \)
Sigma notation: \( \sum_{i=1}^{5} i^2 \)
5. Partial Sums
Definition:
A partial sum is the sum of the first n terms of a series
\[ S_n = a_1 + a_2 + a_3 + \cdots + a_n \]
• \( S_1 = a_1 \) (first partial sum)
• \( S_2 = a_1 + a_2 \) (second partial sum)
• \( S_3 = a_1 + a_2 + a_3 \) (third partial sum)
• And so on...
Formulas:
Arithmetic Partial Sum:
\[ S_n = \frac{n}{2}(a_1 + a_n) \]
Geometric Partial Sum:
\[ S_n = \frac{a_1(1 - r^n)}{1 - r} \]
6. Infinite Geometric Series
Convergence Test:
An infinite geometric series converges (has a finite sum) only if:
\[ |r| < 1 \]
• If \( |r| < 1 \): Series converges (has a sum)
• If \( |r| \geq 1 \): Series diverges (no sum)
Sum Formula (when convergent):
\[ S = \frac{a_1}{1 - r} \quad \text{(for } |r| < 1 \text{)} \]
where \( a_1 \) = first term, \( r \) = common ratio
Examples:
Find sum: \( 8 + 4 + 2 + 1 + \frac{1}{2} + \cdots \)
First term: \( a_1 = 8 \)
Common ratio: \( r = \frac{4}{8} = \frac{1}{2} \)
Since \( |r| = \frac{1}{2} < 1 \), series converges
Use formula: \( S = \frac{a_1}{1 - r} = \frac{8}{1 - \frac{1}{2}} = \frac{8}{\frac{1}{2}} \)
Sum = 16
Does \( 3 + 6 + 12 + 24 + \cdots \) converge?
Common ratio: \( r = \frac{6}{3} = 2 \)
Since \( |r| = 2 > 1 \), series diverges
No sum exists (diverges)
7. Repeating Decimals as Fractions
Method:
Use infinite geometric series to convert repeating decimals to fractions
1. Write the decimal as a geometric series
2. Identify \( a_1 \) and \( r \)
3. Use infinite geometric series formula
4. Simplify the fraction
Examples:
Convert: \( 0.\overline{3} = 0.333\ldots \)
Write as series: \( 0.3 + 0.03 + 0.003 + \cdots \)
Or: \( \frac{3}{10} + \frac{3}{100} + \frac{3}{1000} + \cdots \)
First term: \( a_1 = \frac{3}{10} \)
Common ratio: \( r = \frac{1}{10} \)
Sum: \( S = \frac{a_1}{1-r} = \frac{\frac{3}{10}}{1-\frac{1}{10}} = \frac{\frac{3}{10}}{\frac{9}{10}} = \frac{3}{9} \)
\( 0.\overline{3} = \frac{1}{3} \)
Convert: \( 0.\overline{27} = 0.272727\ldots \)
Write as series: \( 0.27 + 0.0027 + 0.000027 + \cdots \)
Or: \( \frac{27}{100} + \frac{27}{10000} + \frac{27}{1000000} + \cdots \)
First term: \( a_1 = \frac{27}{100} \)
Common ratio: \( r = \frac{1}{100} \)
Sum: \( S = \frac{\frac{27}{100}}{1-\frac{1}{100}} = \frac{\frac{27}{100}}{\frac{99}{100}} = \frac{27}{99} \)
\( 0.\overline{27} = \frac{27}{99} = \frac{3}{11} \)
8. Quick Reference Summary
Key Formulas:
Arithmetic Series: \( S_n = \frac{n}{2}(a_1 + a_n) \) or \( \frac{n}{2}[2a_1 + (n-1)d] \)
Finite Geometric Series: \( S_n = \frac{a_1(1-r^n)}{1-r} \)
Infinite Geometric Series: \( S = \frac{a_1}{1-r} \) (if \( |r| < 1 \))
Sigma Notation: \( \sum_{i=1}^{n} a_i = a_1 + a_2 + \cdots + a_n \)
Convergence: Infinite geometric series converges if \( |r| < 1 \)
📚 Study Tips
✓ Series is the sum of terms in a sequence
✓ Arithmetic: add same value each time (use average method)
✓ Geometric: multiply by same ratio (check if |r| < 1 for infinite series)
✓ Sigma notation is a compact way to write series
✓ Repeating decimals can be written as fractions using infinite geometric series