Sequences and Series
Cheat sheet.

The key rules, formulas and reminders from all 4 topics, gathered into reference cards.

Key formulas, conditions and traps · Read down each column.

Sequences, Geometric and Telescoping Series

Core rule

A series is the limit of its partial sums. $\sum_{n=0}^\infty ar^n=a/(1-r)$ for $|r|<1$ .

Watch for

Terms tending to zero do not prove convergence. Simplify telescoping finite sums before taking limits.

Integral and Comparison Tests

Core rule

Positive $p$ -series converge for $p>1$ . Limit comparison uses $a_n/b_n\to L$ with $0<L<\infty$ .

Watch for

The integral test requires an eventually positive, continuous, decreasing function. Check the direction of direct comparison.

Ratio and Root Tests

Core rule

Ratio or root limit: $L<1$ gives absolute convergence; $L>1$ gives divergence; $L=1$ is inconclusive.

Watch for

Use absolute values. A test that is inconclusive is not a proof of divergence.

Alternating Series and Error Control

Core rule

Decreasing nonnegative $b_n\to0$ implies convergence of $\sum(-1)^nb_n$ , with $|R_N|\le b_{N+1}$ .

Watch for

Absolute and conditional convergence are different. Check monotonicity before using a next-term error bound.

Sequences, Geometric and Telescoping Series

2 reference blocks

Read lesson ↗

Core rule

A series is the limit of its partial sums. $\sum_{n=0}^\infty ar^n=a/(1-r)$ for $|r|<1$ .

Watch for

Terms tending to zero do not prove convergence. Simplify telescoping finite sums before taking limits.

Integral and Comparison Tests

2 reference blocks

Read lesson ↗

Core rule

Positive $p$ -series converge for $p>1$ . Limit comparison uses $a_n/b_n\to L$ with $0<L<\infty$ .

Watch for

The integral test requires an eventually positive, continuous, decreasing function. Check the direction of direct comparison.

Ratio and Root Tests

2 reference blocks

Read lesson ↗

Core rule

Ratio or root limit: $L<1$ gives absolute convergence; $L>1$ gives divergence; $L=1$ is inconclusive.

Watch for

Use absolute values. A test that is inconclusive is not a proof of divergence.

Alternating Series and Error Control

2 reference blocks

Read lesson ↗

Core rule

Decreasing nonnegative $b_n\to0$ implies convergence of $\sum(-1)^nb_n$ , with $|R_N|\le b_{N+1}$ .

Watch for

Absolute and conditional convergence are different. Check monotonicity before using a next-term error bound.