AP Calculus · Unit 10
Infinite Sequences and Series
Convergence tests, Taylor polynomials and power series.
Topic 10.1Sequences, partial sums and convergence
BC only- Distinguish a sequence from a seriesSeparate individual terms, partial sums, and an infinite total.
- Find a sequence limitRecognize convergence, unbounded behavior, or oscillation.
- Use a partial-sum formulaEvaluate partial sums and take their limit to decide series convergence.
- Recover terms or tails from partial sumsUse consecutive partial-sum differences or total minus partial sum.
- Simplify a telescoping sumKeep the uncanceled boundary terms and take the correct limit.
Topic 10.2Geometric series
BC only- Identify a geometric seriesFind a consistent ratio and the actual first included term.
- Check the convergence conditionRequire the magnitude of the ratio to be less than one.
- Find a finite or infinite sumUse the appropriate formula with the correct starting index.
- Use a geometric model in contextTranslate repeated changes into terms and interpret the total.
Topic 10.3A necessary condition for convergence
BC only- Find the limit of the termsEvaluate the nth term as n increases, including oscillating cases.
- Use a nonzero or missing term limitConclude divergence when the terms do not approach zero.
- Recognize the inconclusive zero caseDo not conclude convergence only because the terms approach zero.
- Choose a justified next conclusionDistinguish what the nth-term test proves from what requires another test.
Topic 10.4Integral test
BC only- Check the integral-test conditionsVerify an appropriate positive, continuous, eventually decreasing comparison function.
- Set up the related improper integralMatch the function to the series terms and the appropriate tail.
- Use the integral to decide convergenceTransfer the convergence conclusion without claiming the integral equals the series sum.
- Bound a remaining tailUse lower and upper integral bounds with the correct index offsets.
Topic 10.5Harmonic and p-series
BC only- Recognize a p-seriesRewrite terms to identify the reciprocal-power exponent.
- Apply the p-series thresholdDistinguish convergence for exponents greater than one from divergence otherwise.
- Recognize harmonic divergenceIdentify the exponent-one boundary case.
- Handle constants and finite prefixesRecognize that multiplying by a nonzero constant or changing finitely many terms does not change convergence.
Topic 10.6Comparison tests
BC only- Choose a useful comparison seriesMatch the dominant behavior to a known convergent or divergent series.
- Use direct comparison in the correct directionCheck positivity and use a convergent upper bound or divergent lower bound.
- Carry out a limit comparisonForm the term ratio and interpret a finite positive limit correctly.
- Recognize an inconclusive comparisonDo not reverse an inequality, and do not claim too much from an unsuitable limit.
Topic 10.7Alternating series
BC only- Identify the alternating signs and magnitudesSeparate the sign pattern from the positive term size.
- Check that magnitudes approach zeroEvaluate the necessary limit rather than relying on alternation alone.
- Check eventual decreaseEstablish the monotonicity condition for the alternating-series test.
- State only the supported conclusionDecide when this test proves convergence and when another argument is needed.
Topic 10.8Ratio test
BC only- Form the absolute term ratioUse consecutive terms and include absolute values.
- Simplify the ratioCancel factorial, exponential, and power factors correctly.
- Evaluate and interpret the ratio limitDistinguish a result below one, above one, or equal to one.
- Handle an inconclusive ratio testChoose another suitable test rather than concluding convergence or divergence from a limit of one.
Topic 10.9Absolute versus conditional convergence
BC only- Test the absolute-value seriesDetermine whether the sum of term magnitudes converges.
- Test the original series when necessaryApply the appropriate alternating or other test if absolute convergence fails.
- Classify convergence correctlyDistinguish absolute convergence, conditional convergence, and divergence.
- Connect each conclusion to a valid testCheck test conditions instead of relying on the presence of alternating signs.
Topic 10.10Error of an alternating partial sum
BC only- Check that the alternating error bound appliesVerify decreasing magnitudes tending to zero on the relevant tail.
- Identify the first omitted termUse the correct index after the stated partial sum.
- Bound the size and direction of the errorUse the next magnitude and sign to place the true sum relative to the approximation.
- Choose enough terms for a toleranceSolve the error inequality. Then select an integer number of terms that meets it.
Topic 10.11Taylor polynomial approximation
BC only- Build a Taylor polynomial at the correct centerUse derivative values divided by factorials with powers of the displacement from the center.
- Distinguish coefficients from derivativesConvert between them using the factorial and correct derivative order.
- Use the polynomial to approximate a valueSubstitute the nearby input and retain the requested degree.
- Read the information a polynomial providesDistinguish exact derivative data at the center from an approximation away from it.
Topic 10.12Lagrange remainder
BC only- Choose the derivative order for the remainderUse the derivative one order above the polynomial's degree.
- Find a valid derivative boundBound its magnitude over the entire interval between the center and target.
- Calculate a Lagrange error boundInclude the correct distance power and factorial.
- Use the bound to support an approximationDecide whether the error bound guarantees the requested tolerance. When the question asks, choose a sufficient degree.
Topic 10.13Power-series convergence interval
BC only- Find the center and radius of convergenceDerive the open convergence range, including zero or infinite radius cases.
- Test each finite endpoint separatelySubstitute into the original series and use a suitable convergence test.
- State the complete convergence intervalInclude or exclude each endpoint using correct interval notation.
- Check transformed inputsTranslate the convergence condition back to the original input after a substitution or scaling.
Topic 10.14Build Taylor and Maclaurin series
BC only- Recognize standard seriesIdentify the geometric, exponential, sine, cosine, and logarithmic building blocks used in the course.
- Build a series by substitution or scalingTransform every coefficient and power consistently.
- Write the requested terms or general termKeep the center, starting index, signs, and factorial pattern correct.
- State where the representation is validCarry the appropriate convergence restrictions into the new series.
Topic 10.15Operate on power series
BC only- Differentiate a power series term by termChange coefficients, exponents, and the starting index correctly.
- Integrate a power series term by termInclude the constant and determine it from a known value when needed.
- Combine series to represent a new functionUse supported addition, multiplication, or substitution operations correctly.
- Recheck the convergence intervalPreserve the radius under differentiation or integration but test finite endpoints again.
- Use the resulting seriesFind a coefficient, function value approximation, or definite-integral approximation as requested.