Sequences, series, and convergence
What students will learn
Distinguish terms from partial sums, choose justified convergence tests, and bound the uncertainty in an alternating partial sum.
Before this topic
- Limits at infinity and relative growth
- Positive comparisons and improper integrals
Read and connect
Read: Defining Convergent and Divergent Infinite Series →
Distinguish a term sequence from partial sums and use their finite limit to define a series sum.
Read: Working with Geometric Series →
Use the actual first included term and ratio, verifying convergence before applying a geometric sum.
Read: The nth Term Test for Divergence →
Use a nonzero or missing term limit to prove divergence and recognize that a zero limit is inconclusive.
Read: Integral Test for Convergence →
Verify a positive, continuous, decreasing tail before using an improper integral to decide convergence.
Read: Harmonic Series and p-Series →
Use harmonic and p-series benchmarks with the correct exponent and strict convergence threshold.
Read: Comparison Tests for Convergence →
Choose a positive comparison with the useful inequality direction or a finite positive ratio limit.
Read: Alternating Series Test for Convergence →
Check eventually decreasing magnitudes and a zero limit before using alternating-series convergence.
Read: Ratio Test for Convergence →
Use absolute successive-term ratios and switch tests when the limit is one or does not exist.
Read: Determining Absolute or Conditional Convergence →
Test both signed and magnitude sums to distinguish absolute, conditional, and divergent behavior.
Read: Alternating Series Error Bound →
Use the first omitted alternating magnitude to give a justified error bound and interval for the sum.
Connect: Radius and Interval of Convergence of Power Series →
Later connection: reuse convergence tests for power-series radius and endpoint decisions after introducing power series.
Before this connection: Power-series form and center; Ratio and alternating-series tests.
Check understanding
Use the complete prompt to make student reasoning visible, then consult the separate worked reasoning and criteria. A reading completion tick records reading history.
Repair a zero-term convergence claim →
Explain why a zero term limit is only necessary and use the harmonic benchmark to classify the series.
Reject an inapplicable alternating test →
Check the magnitude behavior, then use finite partial-sum decomposition to make a valid divergence argument.
Trap an alternating sum →
Verify decreasing-to-zero magnitudes, use the first omitted term, and state both an error bound and trapping interval.
Standards and earlier reading lists
These topic placements organize reading. Edition-qualified Standards mappings and the existing complete unit task sets retain their reviewed scope.