Calculus II · Unit 4A · review

Sequence Review

Summary

A sequence converges when its terms approach one finite value. Use dominant terms, algebraic limit laws, squeeze bounds, and monotone-bounded reasoning. Recursive limits require existence before passing to the limit equation.

Exercise

Find the limit of $(5n-2)/(2n+7)$ .

Exercise

Determine whether $(-1)^n+1/n$ converges.

Exercise

Prove $\sin n/n\to0$ .

Exercise

Show that $1-1/n$ is monotone and bounded.

Exercise

Analyze $a_1=2$ , $a_{n+1}=(a_n+5/a_n)/2$ .

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.

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.