Calculus II · Unit 4A · reference

Unit 4A Reference Sheet

{2} Geometric: $\sum ar^n=a/(1-r)$ for $|r|<1$ .

$p$ -series: $\sum1/n^p$ converges iff $p>1$ .

nth term: nonzero term limit implies divergence.

Integral Test: positive, continuous, decreasing interpolation.

Comparison: smaller than convergent; larger than divergent.

Alternating: decreasing magnitudes to zero.

Ratio/Root: limit below one converges absolutely; above one diverges; equal one inconclusive.

Alternating error: at most first omitted magnitude.

After the explanation

Use the section idea

Reading lens

Mixed work removes the method label; classify first, complete an honest attempt, then diagnose the earliest incorrect decision.

Mental model

A complete series argument includes hypotheses, a named theorem or comparison, its conclusion, and any requested sum or error estimate.

Decision

Identify the object and structure before computing, then use answer reveals only after recording a complete justification.

Common trap

Reading the key before choosing a test or treating a missing hypothesis as a minor presentation issue.

Check yourself

Can you reproduce the argument without the key and explain why a tempting alternative test is weaker?

Source & rights

Original instruction with traceable references.

BetterGrades-original; no direct adaptation declared in the verified handoff.

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.