Power series and Taylor approximation
What students will learn
Represent functions with series, build local Taylor polynomials, and justify convergence intervals and approximation error.
Before this topic
- Derivative data and antiderivatives
- Series convergence tests and alternating error bounds
Read and connect
Read: Finding Taylor Polynomial Approximations of Functions →
Build a center-specific Taylor polynomial from derivative values and label its value as an approximation.
Read: Lagrange Error Bound →
Bound the next derivative across the whole interval before stating a Lagrange remainder guarantee.
Read: Radius and Interval of Convergence of Power Series →
Determine radius and open convergence interval, then test each endpoint of the original series separately.
Read: Finding Taylor or Maclaurin Series for a Function →
Build Taylor or Maclaurin series and justify function representation with a vanishing remainder.
Read: Representing Functions as Power Series →
Derive function representations by valid series operations, fixing constants and rechecking endpoints.
Connect: Ratio Test for Convergence →
Revisit the ratio test to find an open power-series convergence interval before testing endpoints.
Before this connection: Series convergence and alternating hypotheses.
Connect: Alternating Series Error Bound →
Revisit first-omitted-term bounds when a Taylor or derived series is alternating.
Before this connection: Series convergence and alternating hypotheses.
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.
Justify a Taylor approximation’s error bound →
Build the degree-two polynomial and apply the supplied derivative bound over the entire center-to-target interval.
Test both convergence endpoints →
Determine the radius from a ratio inequality, then classify each endpoint in the original series.
Uses one given power series with a finite positive radius. It does not ask students to state the general definition or decide center-only or infinite-radius cases.
Derive and check an integrated series →
Integrate a geometric series, fix its constant, justify endpoint representation, and compare a four-term approximation with its alternating error bound.
The task derives an integrated series and interval; it does not explicitly ask students to compare its radius with the starting series.
Compare center-only, finite, and infinite radii →
Evaluate the center separately, decide each convergence set, and draw number lines with explicit boundary inclusion.
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.