![]() ![]() Your teacher may decide to teach you about more tests, but these are the 6 that are tested on the AP exam.ġ. It is essential that you memorize these 6 tests, their conditions, and their outcomes for the AP exam. It is also used to approximate functions that are difficult to integrate or differentiate. The Maclaurin series can be used to find the values of a function and its derivatives at 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. Maclaurin Series: The Maclaurin series is a special case of the Taylor series when the center of expansion is 0. It is used to find an approximation of a function, especially when the function cannot be expressed in closed form.ħ. 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. 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.Ħ. Alternating Series: An alternating series is a series in which the terms alternate in sign. The radius of convergence of a power series is the distance from c within which the series converges.ĥ. 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 nth term of the Harmonic series is given by 1/n. ![]() Harmonic Series: The Harmonic series is a series in which the terms are the reciprocals of the positive integers. The nth term of a geometric sequence is given by a*r^(n-1).ģ. It can be represented as a, ar, ar^2, ar^3, where a is the first term and r is the common ratio. Geometric Sequence: A geometric sequence is a sequence in which each term is equal to the previous term multiplied by a fixed constant. The nth term of an arithmetic sequence is given by a+(n-1)d.Ģ. It can be represented as a, a+d, a+2d, a+3d, where a is the first term and d is the common difference. Arithmetic Sequence: An arithmetic sequence is a sequence in which the difference between any two consecutive terms is a constant. In this unit, you will learn how to use a variety of tests to determine whether a sequence or series converges or diverges.ġ. 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. A convergent sequence or series is one in which the terms of the sequence or series approach a specific value, called the limit. There are two types of sequences and series: convergent and divergent. 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. Sequences and series are used to study the behavior of infinite sums of numbers. This means that it is up to you to find the one to use to fit that series. There are many tests that can be used to determine series convergence and divergence. This means that you know how to rewrite any function as an infinite power series.Ĥ) Apply an appropriate mathematical definition, theorem, or test. 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.ģ) Identify a re-expression of mathematical information presented in a given representation. 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.Ģ) Explain how an approximated value relates to the actual value. Mathematical Practicesįour of the College Board's mathematical practices for AP Calculus are used in this unit, which are outlined below.ġ) Apply appropriate mathematical rules or procedures, with and without technology. The concepts in this unit may seem difficult at first, but with a little practice, you will be able to understand them. You will also need to understand the two types of error bounds you will encounter in this unit. It is important that you know the conditions for all the tests and what they tell you. You will learn 6 tests for different types of series to glean information about them. 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. This unit emphasizes the study of series analysis. This unit builds from past knowledge of limits, differentiation, and integration, but applies this to a new concept - series. 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! Always one FRQ on this unit (usually FRQ 6) ![]()
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