Calculus II · Unit 4A · practice

Convergence-Test Selection Practice

Practice method

Work in three passes

First, classify. Name the derivative idea or rule before writing algebra. This separates a recognition error from a calculation error.

Second, solve without the key. Record a complete attempt, including bounds, constants, domains, units, or interpretation when the prompt asks for them.

Third, reveal one answer at a time. Compare the first line where your work differs, close the answer, and redo that item from a blank start.

Convergence-Test Selection Practice

For each series, name the first test you would try and explain why before performing calculations.

Exercise

$\sum (n+1)/(n^3+2)$

Exercise

$\sum (-1)^n n/(n^2+1)$

Exercise

$\sum 5^n/n!$

Exercise

$\sum 1/[n\ln n]$

Exercise

$\sum ((2n+3)/(5n+1))^n$

Exercise

$\sum 1/[n(n+2)]$

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?

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