AP Precalculus: Series

Master arithmetic and geometric series formulas, sigma notation, and infinite series

∑ Sigma Notation ➕ Arithmetic Series ✖️ Geometric Series ∞ Infinite Series

📚 Understanding Series

A series is the sum of terms in a sequence. While a sequence lists numbers, a series adds them up. AP Precalculus focuses on arithmetic series (constant difference), geometric series (constant ratio), and the powerful sigma notation used to express sums compactly.

1 Series & Sigma Notation

A series is the sum of terms in a sequence. Sigma notation (Σ) provides a compact way to write sums.

\(\displaystyle\sum_{k=m}^{n} a_k = a_m + a_{m+1} + a_{m+2} + \cdots + a_n\)

Sum of \(a_k\) as \(k\) goes from \(m\) to \(n\)

Index variable: \(k\) (or \(i\), \(n\)) — the variable that changes
Lower limit: \(m\) — starting value of the index
Upper limit: \(n\) — ending value of the index
General term: \(a_k\) — the expression being summed

📌 Example

\(\displaystyle\sum_{k=1}^{5} k^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 1 + 4 + 9 + 16 + 25 = 55\)

2 Arithmetic Series

An arithmetic series is the sum of terms in an arithmetic sequence (constant difference \(d\) between terms).

Formula 1 (First & Last)

\(S_n = \frac{n}{2}(a_1 + a_n)\)

Use when you know first and last terms

Formula 2 (First & Difference)

\(S_n = \frac{n}{2}[2a_1 + (n-1)d]\)

Use when you know first term and common difference

where \(n\) = number of terms, \(a_1\) = first term, \(a_n\) = last term, \(d\) = common difference

📌 Example

Find the sum: \(3 + 7 + 11 + 15 + 19 + 23\)

Identify: \(a_1 = 3\), \(a_n = 23\), \(d = 4\), \(n = 6\)

Using Formula 1: \(S_6 = \frac{6}{2}(3 + 23) = 3(26) = 78\)

Using Formula 2: \(S_6 = \frac{6}{2}[2(3) + (5)(4)] = 3[6 + 20] = 3(26) = 78\) ✓

💡 Memory Trick

Formula 1 is simply: (number of terms) × (average of first and last terms)

3 Geometric Series (Partial Sum)

A geometric series is the sum of terms in a geometric sequence (constant ratio \(r\) between terms).

Finite Geometric Series (r ≠ 1) \(S_n = a_1 \cdot \frac{1 - r^n}{1 - r}\)

where \(n\) = number of terms, \(a_1\) = first term, \(r\) = common ratio

Alternative Form

\(S_n = a_1 \cdot \frac{r^n - 1}{r - 1}\) (same formula, different arrangement)

When r = 1

\(S_n = n \cdot a_1\) (all terms are equal)

📌 Example

Find the sum: \(2 + 6 + 18 + 54 + 162\)

Identify: \(a_1 = 2\), \(r = 3\), \(n = 5\)

Calculate: \(S_5 = 2 \cdot \frac{1 - 3^5}{1 - 3} = 2 \cdot \frac{1 - 243}{-2} = 2 \cdot \frac{-242}{-2} = 2 \cdot 121 = 242\)

4 Infinite Geometric Series

An infinite geometric series has infinitely many terms. It only has a finite sum if it converges.

Converges ✓

\(|r| < 1\)

Diverges ✗

\(|r| \geq 1\)

Infinite Geometric Series Sum (when |r| < 1) \(S = \frac{a_1}{1 - r}\)

📌 Example

Find the sum: \(8 + 4 + 2 + 1 + \frac{1}{2} + \cdots\)

Identify: \(a_1 = 8\), \(r = \frac{1}{2}\)

Check: \(|r| = \frac{1}{2} < 1\) → converges

Sum: \(S = \frac{8}{1 - \frac{1}{2}} = \frac{8}{\frac{1}{2}} = 16\)

⚠️ Divergent Series

If \(|r| \geq 1\), the series diverges (no finite sum). Example: \(3 + 6 + 12 + 24 + \cdots\) with \(r = 2\) has no sum.

5 Comparing Arithmetic & Geometric Series

Feature	Arithmetic Series	Geometric Series
Pattern	Constant difference \(d\)	Constant ratio \(r\)
nth term	\(a_n = a_1 + (n-1)d\)	\(a_n = a_1 \cdot r^{n-1}\)
Partial Sum	\(S_n = \frac{n}{2}(a_1 + a_n)\)	\(S_n = a_1 \cdot \frac{1-r^n}{1-r}\)
Infinite Sum	Always diverges	Converges if \(\|r\| < 1\)
Graph of partial sums	Quadratic growth	Exponential growth/decay

6 Repeating Decimals as Fractions

A repeating decimal can be written as an infinite geometric series, then converted to a fraction using the infinite sum formula.

Step 1: Write as Series

Express the repeating decimal as sum of terms

Step 2: Identify a₁ and r

First term and common ratio (usually power of 0.1)

Step 3: Apply Formula

Use \(S = \frac{a_1}{1-r}\)

Step 4: Simplify

Reduce the fraction to lowest terms

📌 Example: Convert \(0.\overline{27}\) to a fraction

Write as series: \(0.27 + 0.0027 + 0.000027 + \cdots\)

Identify: \(a_1 = 0.27 = \frac{27}{100}\), \(r = 0.01 = \frac{1}{100}\)

Apply formula: \(S = \frac{\frac{27}{100}}{1 - \frac{1}{100}} = \frac{\frac{27}{100}}{\frac{99}{100}} = \frac{27}{99} = \frac{3}{11}\)

💡 Quick Method

For \(0.\overline{ab}\): write as \(\frac{ab}{99}\). For \(0.\overline{abc}\): write as \(\frac{abc}{999}\). Then simplify!

7 Sigma Notation Properties

These properties help simplify and evaluate sums in sigma notation.

Constant multiple: \(\sum_{k=1}^{n} c \cdot a_k = c \sum_{k=1}^{n} a_k\)
Sum of sums: \(\sum_{k=1}^{n} (a_k + b_k) = \sum_{k=1}^{n} a_k + \sum_{k=1}^{n} b_k\)
Constant sum: \(\sum_{k=1}^{n} c = n \cdot c\)
Sum of first n integers: \(\sum_{k=1}^{n} k = \frac{n(n+1)}{2}\)
Sum of first n squares: \(\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}\)

📋 Quick Reference

Arithmetic Sum

\(S_n = \frac{n}{2}(a_1 + a_n)\)

Geometric Sum

\(S_n = a_1 \cdot \frac{1-r^n}{1-r}\)

Infinite Geometric

\(S = \frac{a_1}{1-r}\) if \(|r| < 1\)

Sum of Integers

\(\sum k = \frac{n(n+1)}{2}\)

Convergence Test

\(|r| < 1\) → converges

Repeating Decimal

\(0.\overline{ab} = \frac{ab}{99}\)

📚 Understanding Series

1 Series & Sigma Notation

2 Arithmetic Series

3 Geometric Series (Partial Sum)

Alternative Form

When r = 1

4 Infinite Geometric Series

5 Comparing Arithmetic & Geometric Series

6 Repeating Decimals as Fractions

Step 1: Write as Series

Step 2: Identify a₁ and r

Step 3: Apply Formula

Step 4: Simplify

7 Sigma Notation Properties

📋 Quick Reference

Arithmetic Sum

Geometric Sum

Infinite Geometric

Sum of Integers

Convergence Test

Repeating Decimal

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