Well, after going through countless limits, derivatives, and integrals, you’ve made it here! Welcome to Unit 10, the last unit of AP Calculus BC! Before we move on, give yourself a pat on the back for all your hard work!
This unit builds from past knowledge of limits, differentiation, and integration, but applies this to a new concept — series. This unit emphasizes the study of series analysis. If you haven’t ever heard of analysis in a mathematical sense before, this is just the study of functions and how they behave over time.
You will learn 6 tests for different types of series to glean information about them. It is important that you know the conditions for all the tests and what they tell you. You will also need to understand the two types of error bounds you will encounter in this unit. The concepts in this unit may seem difficult at first, but with a little practice, you will be able to understand them.
Four of the College Board's mathematical practices for AP Calculus are used in this unit, which are outlined below.
1) Apply appropriate mathematical rules or procedures, with and without technology.
This means that you should know how to calculate the error bounds to analyze series, with the alternating error bound for alternating series and the Lagrange Error Bound for power series.
2) Explain how an approximated value relates to the actual value.
This means that you know how to interpret Lagrange and alternating series error bounds as the maximum error of an infinite series given a partial series.
3) Identify a re-expression of mathematical information presented in a given representation.
This means that you know how to rewrite any function as an infinite power series.
4) Apply an appropriate mathematical definition, theorem, or test.
There are many tests that can be used to determine series convergence and divergence. This means that it is up to you to find the one to use to fit that series.
Limit Comparison Test
Direct Comparison Test
Alternating Series Test
Sequences and series are used to study the behavior of infinite sums of numbers. A sequence is a function whose domain is the set of natural numbers, and a series is the sum of the terms of a sequence.
There are two types of sequences and series: convergent and divergent. A convergent sequence or series is one in which the terms of the sequence or series approach a specific value, called the limit. A divergent sequence or series is one in which the terms do not approach a specific value and therefore, the sum of the terms is infinite. In this unit, you will learn how to use a variety of tests to determine whether a sequence or series converges or diverges.
1. Arithmetic Sequence: An arithmetic sequence is a sequence in which the difference between any two consecutive terms is a constant. It can be represented as a, a+d, a+2d, a+3d, where a is the first term and d is the common difference. The nth term of an arithmetic sequence is given by a+(n-1)d.
2. Geometric Sequence: A geometric sequence is a sequence in which each term is equal to the previous term multiplied by a fixed constant. It can be represented as a, ar, ar^2, ar^3, where a is the first term and r is the common ratio. The nth term of a geometric sequence is given by a*r^(n-1).
3. Harmonic Series: The Harmonic series is a series in which the terms are the reciprocals of the positive integers. The nth term of the Harmonic series is given by 1/n. This series diverges.
4. Power Series: A power series is a series of the form ∑a_n(x-c)^n where a_n are constants and c is a constant. The radius of convergence of a power series is the distance from c within which the series converges.
5. Alternating Series: An alternating series is a series in which the terms alternate in sign. Such as a_1 - a_2 + a_3 - a_4 + .... The terms of an alternating series must decrease in absolute value and the limit of the terms must be zero for the series to converge.
6. Taylor Series: The Taylor series is a representation of a function as an infinite sum of terms, with each term being a polynomial function of a single variable. It is used to find an approximation of a function, especially when the function cannot be expressed in closed form.
7. Maclaurin Series: The Maclaurin series is a special case of the Taylor series when the center of expansion is 0. It is a representation of a function as an infinite sum of terms, with each term being a polynomial function of a single variable. The Maclaurin series can be used to find the values of a function and its derivatives at 0. It is also used to approximate functions that are difficult to integrate or differentiate.
It is essential that you memorize these 6 tests, their conditions, and their outcomes for the AP exam. Your teacher may decide to teach you about more tests, but these are the 6 that are tested on the AP exam.
1. nth Term Test/Limit Test: This test is used to determine if a series diverges by finding the limit of the nth term of the series as n approaches infinity. If the limit is non-zero or does not exist, the series diverges. If the limit is zero, you cannot conclude anything from this test (it is inconclusive), and you will need to select another test to investigate further. To perform this test, take the nth term of the series and find its limit as n approaches infinity.
2. Limit Comparison Test: Limit Comparison test is used to compare the ratio of two series, and if the limit of the ratio is finite and positive, the two series will both either converge or diverge. However, the two series may not converge to the exact value of the limit.
3. Direct Comparison Test: This test is used to compare the absolute value of the terms of a series with the absolute value of the terms of another series that has already been proven to converge or diverge. If the larger series converges, the smaller series converges, and if the smaller series diverges, the bigger series diverges. The two statements do not work the other way around.
4. Integral Test: This test is used to determine the convergence of a series by comparing it to an integral. To perform this test, you need to find an integral that has the same terms as the series and check whether this integral converges or diverges. If the integral converges, the series converges, and if it diverges, the series diverges.
5. Alternating Series Test: This test is used for series whose terms alternate in sign. To perform this test, you need to check if the terms decrease in absolute value and if the limit of the terms is zero. If these two conditions are met, the series converges.
6. Ratio Test: This test is used to determine the convergence of a series by finding the limit of the ratio of consecutive terms. To perform this test, take the ratio of consecutive terms in the series and find its limit as n approaches infinity. If the limit is less than 1, the series converges absolutely. If the limit is greater than 1, the series diverges. If the limit is equal to 1, the test is inconclusive.
The Lagrange Error Bound is a bound on the error made when approximating a definite integral using a polynomial function. It is based on the fact that the polynomial of degree n that best approximates a function f(x) on an interval [a,b] is the n-th degree Taylor polynomial of f(x) centered at some point c in that interval.
The Alternating Series Error Bound is a bound on the error made when approximating a series. It is used for alternating series which are series in which the terms alternate in sign. It states that if the absolute value of the series is decreasing and the limit of the terms is zero, then the error made in approximating the series by the first n terms is less than or equal to the absolute value of the (n+1)th term.
In addition to the series, tests, and error bounds, you will also learn how to find the radius and interval of convergence of power series, and how to find the power series representation of a series. You will also be taking derivatives and integrals of series term-by-term. This unit contains a lot of content, but as you read through the guides for this unit and practice the concepts, you will start to see the connections in between topics. And once you're done with this unit, you will be done with all of the content for AP Calculus BC! You're almost there, so finish strong!