Calculus II · Unit 4A · reference

Unit 4A Reference Sheet

Use Unit 4A Reference Sheet as a concise reference for formulas, theorem hypotheses, decisions, and common errors.

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Section overview

Review, practice, exams, and answer keys

What this section is building

Use Unit 4A Reference Sheet as a concise reference for formulas, theorem hypotheses, decisions, and common errors.

Notice

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

Decide

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

Avoid

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

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.

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.

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.

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.