Sequence and Series Cheatsheet

Cheat Codes & Shortcuts

• Sequence: Ordered list of numbers \( \{a_n\} \), e.g., \( a_n = n^2 \).
• Series: Sum of sequence terms \( \sum_{n=1}^\infty a_n \).
• Arithmetic Sequence: \( a_n = a_1 + (n-1)d \), where \( d \) is common difference.
• Geometric Sequence: \( a_n = a_1 r^{n-1} \), where \( r \) is common ratio.
• Arithmetic Series Sum: \( S_n = \frac{n}{2} (a_1 + a_n) \).
• Geometric Series Sum: \( S_n = a_1 \frac{1 - r^n}{1 - r} \) (finite), \( S_\infty = \frac{a_1}{1 - r} \), \( |r| < 1 \).
• Convergence: Series converges if \( \lim_{n \to \infty} a_n = 0 \) (necessary but not sufficient).
• Test for Divergence: If \( \lim_{n \to \infty} a_n \neq 0 \), series diverges.
• Ratio Test: For \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \): \( < 1 \) converges, \( > 1 \) diverges.
• Integral Test: For \( f(n) = a_n \), if \( \int_1^\infty f(x) \, dx \) converges, so does series.

Quick Reference Table

Type	Form	Solution/Method
Arithmetic	\( a_n = 3 + 2(n-1) \)	Sequence: \( 3, 5, 7, \ldots \); Sum: \( S_n = \frac{n}{2} (3 + a_n) \)
Geometric	\( a_n = 2 \cdot 3^{n-1} \)	Sum: \( S_n = 2 \frac{1 - 3^n}{1 - 3} \)
Infinite Geometric	\( \sum_{n=1}^\infty \frac{1}{2^n} \)	Sum: \( \frac{1/2}{1 - 1/2} = 1 \)
Ratio Test	\( \sum_{n=1}^\infty \frac{n}{2^n} \)	Compute \( \lim_{n \to \infty} \left| \frac{n+1}{2^{n+1}} \cdot \frac{2^n}{n} \right| \)
Integral Test	\( \sum_{n=1}^\infty \frac{1}{n^2} \)	Evaluate \( \int_1^\infty \frac{1}{x^2} \, dx \)
P-Series	\( \sum_{n=1}^\infty \frac{1}{n^p} \)	Converges if \( p > 1 \), diverges if \( p \leq 1 \)

Advice

Sequence and Series Quick Tips

• Arithmetic Sequence: Check for constant difference: \( a_{n+1} - a_n = d \).
• Geometric Sequence: Check for constant ratio: \( \frac{a_{n+1}}{a_n} = r \).
• Series Convergence: Use Ratio or Integral Test for complex series.
• P-Series: \( \sum \frac{1}{n^p} \) converges for \( p > 1 \).
• Test for Divergence: If \( \lim_{n \to \infty} a_n \neq 0 \), series diverges.

Sequence and Series Speed Quiz

Test your speed with 5 sequence and series questions! You have 30 seconds per question.