Mixed Review: Infinite Sequences and Series

The Big Picture

Infinite sequences and series are one of the most heavily connected topics in AP Calculus BC. A problem may begin with a sequence, ask whether a series converges, require a convergence test, turn into a power series, or finish by using a Taylor polynomial to approximate a value.

The key is to identify what kind of object you are looking at before choosing a technique. A sequence is a list of terms. A series is an infinite sum of those terms. A power series is a series whose terms contain powers of a variable.

Once you can distinguish those three objects, most mixed-review questions become a matter of selecting the correct tool.

1. Infinite Sequences

A sequence is a function whose domain is the positive integers. We usually write its terms as .

For a sequence, the central question is usually whether the terms approach a finite number as . If they do, the sequence converges. If they do not, the sequence diverges.

Example: Rational Functions of n

2. Infinite Series

A series is an infinite sum. If is a sequence, then the associated series is .

The most important distinction in this entire unit is this: the fact that the terms of a sequence approach zero does not guarantee that the series converges.

For a series to converge, its terms must approach zero, but that is only a necessary condition. The harmonic series is the classic counterexample:

Its individual terms approach zero, but the infinite sum still diverges.

3. The nth-Term Test for Divergence

The fastest way to prove that a series diverges is often the nth-term test. If the terms of the series do not approach zero, the series cannot converge.

4. Geometric Series

A geometric series has a constant ratio between consecutive terms:

The geometric series converges exactly when the absolute value of its common ratio is less than 1.

Example: Recognizing a Geometric Series

5. Choosing a Convergence Test

When a series is not obviously geometric, the biggest AP Calculus BC skill is choosing an appropriate convergence test.

6. p-Series

The p-series is one of the most important benchmarks for comparison.

The entire decision comes down to the exponent:

• If , the series converges.
• If , the series diverges.

7. Comparison Tests

Comparison tests are useful when a complicated series behaves like a simpler series whose convergence is already known.

For positive-term series, direct comparison can be summarized as follows:

The direction matters. A smaller series than a known convergent series must also converge. A larger series than a known divergent series must also diverge.

8. Alternating Series

An alternating series changes sign from one term to the next. A common form is

The alternating series test says that the series converges if the positive magnitudes decrease eventually and approach zero.

9. Absolute Versus Conditional Convergence

A series can converge because its terms alternate even though the corresponding series of absolute values diverges.

If the series of absolute values converges, the original series is absolutely convergent. If the original series converges but the absolute-value series diverges, it is conditionally convergent.

10. The Ratio Test

The ratio test is especially useful when terms contain factorials or expressions raised to the nth power.

If , the series converges absolutely. If , the series diverges. If the limit equals 1, the test is inconclusive.

11. Power Series

A power series is an infinite series involving powers of a variable:

For a power series, the central questions are:

• What is the radius of convergence?
• What is the interval of convergence?
• What happens at each endpoint?

The ratio test often gives the radius of convergence quickly.

12. Manipulating Known Power Series

You do not always need to build a Taylor series from scratch. AP Calculus BC frequently gives you a familiar series and asks you to manipulate it.

The geometric series is the starting point for many important series.

For example, replacing with gives a series for .

Differentiating and integrating known power series is another major technique.

13. Taylor and Maclaurin Series

A Taylor series represents a function as an infinite polynomial centered at a point .

When , the Taylor series becomes a Maclaurin series:

The first few terms come directly from the derivatives of the function at the center.

14. Common Maclaurin Series to Know

Several functions have especially useful Maclaurin series. Recognizing these patterns can save substantial time.

15. Taylor Polynomial Approximation

A Taylor polynomial uses only finitely many terms of a Taylor series. If is the degree-n Taylor polynomial centered at , then it approximates the original function near the center.

16. Lagrange Error Bound

Taylor polynomials are approximations, so AP Calculus BC may ask you to bound the error. The Lagrange error bound gives:

Here is a bound on the absolute value of the -th derivative throughout the interval between the center and the point being approximated.

17. Alternating Series Error Bound

For an alternating series satisfying the conditions of the alternating series test, the error after stopping at a particular term is no larger than the magnitude of the first omitted term.

This is often much easier to use than the Lagrange error bound when the problem already involves an alternating series.

18. A Mixed AP Calculus BC Strategy

When a problem mixes several ideas, do not immediately start calculating. First classify the object and determine what the question is actually asking.

Common Mistakes

Practice Problems

Final AP Exam Checklist

Before moving on from infinite sequences and series, make sure you can answer each of these questions without hesitation:

• Can I distinguish a sequence from a series?
• Can I use the nth-term test to prove divergence?
• Can I recognize and sum a geometric series?
• Can I recognize a p-series?
• Can I choose between comparison, ratio, integral, and alternating-series tests?
• Can I distinguish absolute from conditional convergence?
• Can I find the radius and interval of convergence of a power series?
• Can I test power-series endpoints separately?
• Can I manipulate known power series?
• Can I construct Taylor and Maclaurin series?
• Can I use Taylor polynomials to approximate function values?
• Can I apply the Lagrange error bound?
• Can I use the alternating-series error bound?

If you can move comfortably between these ideas, you are prepared for the mixed series questions that combine several AP Calculus BC skills in a single problem.