Calculus II · Unit 4A · review

Convergence Tests Review

Decision

Ask what structure is visible

Geometric ratio? Telescope? Nonzero term limit? Positive comparison scale? Alternating cancellation? Factorials? An entire expression to the $n$ th power? Choose the least complicated test that fully answers the question.

Exercise

Classify $\sum 4(2/5)^n$ .

Exercise

Classify $\sum1/(n^2+4)$ .

Exercise

Classify $\sum(-1)^n/\sqrt n$ as absolute, conditional, or divergent.

Exercise

Classify $\sum n!/10^n$ .

Exercise

Classify $\sum[(3n+1)/(4n-2)]^n$ .

Exercise

Classify $\sum1/[n(\ln n)^2]$ .

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.

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