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Unit 10: Infinite Sequences and Series

15 dot points across 15 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do you bound the error when approximating an alternating series by a partial sum?

Topic 10.10 Alternating Series Error Bound: bound the error of a partial-sum approximation of an alternating series by the magnitude of the first omitted term (BC).
A focused answer to AP Calculus BC Topic 10.10, bounding the error of approximating a convergent alternating series by a partial sum using the magnitude of the first omitted term, with worked examples.
8 min answer →

How does the alternating series test establish convergence of a series whose signs alternate?

Topic 10.7 Alternating Series Test for Convergence: use the alternating series test (terms decreasing in magnitude to zero) to conclude convergence (BC).
A focused answer to AP Calculus BC Topic 10.7, using the alternating series test (terms decreasing in absolute value to zero) to conclude convergence of an alternating series, with worked examples.
8 min answer →

What does it mean for an infinite series to converge, in terms of its partial sums?

Topic 10.1 Defining Convergent and Divergent Infinite Series: define the convergence of an infinite series as the limit of its sequence of partial sums (BC).
A focused answer to AP Calculus BC Topic 10.1, defining the convergence and divergence of an infinite series through the limit of its sequence of partial sums, distinguishing a sequence from a series, with worked examples.
9 min answer →

What is the difference between absolute and conditional convergence?

Topic 10.9 Determining Absolute or Conditional Convergence: classify a convergent series as absolutely or conditionally convergent by testing the series of absolute values (BC).
A focused answer to AP Calculus BC Topic 10.9, classifying a convergent series as absolutely or conditionally convergent by testing the series of absolute values, with worked examples.
8 min answer →

How do you find the Taylor or Maclaurin series of a function, and what are the standard ones?

Topic 10.14 Finding Taylor or Maclaurin Series for a Function: write the full Taylor or Maclaurin series of a function and recall the standard series for e^x, sin x, cos x and 1/(1-x) (BC).
A focused answer to AP Calculus BC Topic 10.14, writing the full Taylor or Maclaurin series of a function from its derivatives and recalling the standard series for e^x, sin x, cos x and the geometric series, with worked examples.
10 min answer →

How do you build a Taylor polynomial that approximates a function near a point?

Topic 10.11 Finding Taylor Polynomial Approximations of Functions: construct the Taylor (or Maclaurin) polynomial of a function about a center using its derivatives at that point (BC).
A focused answer to AP Calculus BC Topic 10.11, building Taylor and Maclaurin polynomial approximations of a function from its derivatives at the center, and using them to estimate values, with worked examples.
10 min answer →

When does a p-series converge, and why does the harmonic series diverge?

Topic 10.5 Harmonic Series and p-Series: apply the p-series rule (converges iff p greater than 1) and recognize the harmonic series as the divergent p = 1 case (BC).
A focused answer to AP Calculus BC Topic 10.5, applying the p-series convergence rule (converges iff p is greater than 1), recognizing the harmonic series as the divergent borderline case, with worked examples.
8 min answer →

How do you bound the error of a Taylor polynomial approximation using the Lagrange error bound?

Topic 10.12 Lagrange Error Bound: bound the error of a Taylor polynomial approximation using the Lagrange form of the remainder (BC).
A focused answer to AP Calculus BC Topic 10.12, bounding the error of a Taylor polynomial approximation with the Lagrange form of the remainder, using a bound on the next derivative, with worked examples.
9 min answer →

How do you find the radius and interval of convergence of a power series?

Topic 10.13 Radius and Interval of Convergence of Power Series: find the radius and interval of convergence of a power series using the ratio test and checking the endpoints (BC).
A focused answer to AP Calculus BC Topic 10.13, finding the radius and interval of convergence of a power series with the ratio test and separately testing the endpoints, with worked examples.
10 min answer →

How do you manipulate power series to represent new functions and evaluate hard integrals?

Topic 10.15 Representing Functions as Power Series: manipulate known power series by substitution, multiplication, differentiation and integration to represent new functions and evaluate otherwise intractable integrals (BC).
A focused answer to AP Calculus BC Topic 10.15, manipulating known power series by substitution, term-by-term differentiation and integration to represent new functions and evaluate integrals with no elementary antiderivative, with worked examples.
10 min answer →

How do you decide convergence by comparing a series to a known benchmark series?

Topic 10.6 Comparison Tests for Convergence: use the direct comparison test and the limit comparison test against a known p-series or geometric series (BC).
A focused answer to AP Calculus BC Topic 10.6, deciding convergence with the direct comparison test and the limit comparison test against a known p-series or geometric benchmark, with worked examples.
9 min answer →

How does an improper integral decide the convergence of a series with positive decreasing terms?

Topic 10.4 Integral Test for Convergence: use the convergence of a related improper integral to decide convergence of a series with positive, decreasing terms (BC).
A focused answer to AP Calculus BC Topic 10.4, using the integral test to decide convergence of a series with positive decreasing terms by evaluating a related improper integral, with worked examples.
9 min answer →

How does the nth term test show a series diverges, and why can it never prove convergence?

Topic 10.3 The nth Term Test for Divergence: use the limit of the terms to conclude divergence when the terms do not approach zero (BC).
A focused answer to AP Calculus BC Topic 10.3, using the nth term test to conclude divergence when the terms fail to approach zero, and understanding why it can never prove convergence, with worked examples.
8 min answer →

How does the ratio test use the limit of consecutive-term ratios to decide convergence?

Topic 10.8 Ratio Test for Convergence: use the limit of the ratio of consecutive terms to decide absolute convergence or divergence, especially for factorials and exponentials (BC).
A focused answer to AP Calculus BC Topic 10.8, using the ratio test to decide absolute convergence or divergence from the limit of consecutive-term ratios, especially for series with factorials and exponentials, with worked examples.
9 min answer →

When does a geometric series converge, and what is its sum?

Topic 10.2 Working with Geometric Series: determine convergence of a geometric series by its common ratio and find its sum with the a over one-minus-r formula (BC).
A focused answer to AP Calculus BC Topic 10.2, deciding convergence of a geometric series from its common ratio and computing its sum with the a over one-minus-r formula, with worked examples.
9 min answer →