![]() Converges by alternating series or ratio test or rewriting as geometric series. Diverges by limit comparison with harmonic series. Converges by either direct or limit comparison with f 1 2 1 n n 10. 11.6: Comparison Test As we begin to compile a list of convergent and divergent series, new ones can sometimes be analyzed by comparing them to ones that we already understand. INFINITE SEQUENCES AND SERIES ANSWER KEY Chapter 12A Review Sheet 2 1.11.5: Alternating Series Next we consider series with both positive and negative terms, but in a regular pattern: they alternate.This means that if we can show that the sequence of partial sums is bounded, the series must converge. If all of the terms anan in a series are non-negative, then clearly the sequence of partial sums snsn is non-decreasing. In many cases it is possible at least to determine whether or not the series converges, and so we will spend most of our time on this problem. 11.4: The Integral Test It is generally quite difficult, often impossible, to determine the value of a series exactly.A series converges if the sequence of partial sums converges, and otherwise the series diverges. Associated with a series is a second sequence, called the sequence of partial sums. 11.3: Series Recall that a series, roughly speaking, is the sum of a sequence.The range of the function is still allowed to be the real numbers in symbols, we say that a sequence is a function f:N→R. “AP Calculus AB and BC Course and Exam Description, Effective Fall 2019,” The College Board, 2019. Limits and Continuity About 47 of the questions on your exam will cover Limits and Continuity. Calculus - Early Transcendentals, 8th Edition. Infinite Sequences and Series: 1718 of test questions This guide offers an overview of the main tested subjects, along with sample AP multiple-choice questions that look like the questions you’ll see on test day. ![]() No matter what path you choose, remember that calculus, and more generally mathematics is all around us, and keep on studying and remember to make your exams all Fiveable! Geometry is the study of structures and objects in different dimensions, and topology is the study of structures and their properties under different transformations. Geometry and topology are the study of shape and are two related fields. Number theory, which deals with the properties of numbers, is also a subset of algebra as well. Abstract algebra then includes more abstract structures, such as groups, rings, and fields. Linear algebra is usually the first algebra class one will take and will involve the study of matrices and vectors, and introduces the topic of a vector space (which includes more than just vectors). It covers both ordinary differential equations, differential equations with respect to one variable, and partial differential equations, differential equations with respect to multiple variables.Īnalysis is the study of change, especially relating to functions and other curves, and this serves as a theoretical extension to calculus, with real analysis covering real-valued functions and complex analysis covering complex numbers and complex-valued functions.Īlgebra is the study of structure (not to be confused with high school algebra). ![]() Multivariable calculus applies the concepts of single-variable into multiple dimensions and also expands the concept of vector calculus, which eventually sums up and generalizes most of the important theorems in calculus, especially the Fundamental Theorem of Calculus.ĭifferential equations is all about finding ways to solve many different types of differential equations, especially those which are highly applicable to other disciplines. After this, there are multiple courses which follow, which include multivariable calculus, differential equations, real and complex analysis, linear and abstract algebra,, and geometry and topology. ![]() However, there are some concepts in single-variable calculus that aren’t covered in AP Calculus that are useful for higher math courses such as hyperbolic functions, the epsilon-delta definition of a limit, and trigonometric substitutions. If you want to go into higher STEM courses, then you may consider higher mathematical courses. However, AP Physics C uses calculus freely and is a good challenge if you are interested in physics or want a little bit of a challenge! ![]() AP Chemistry utilizes calculus indirectly with the concepts of reaction rates and rate laws. Many science courses are also deeply rooted in calculus as well. A lot of statistics is based off of calculus, even if the relationship isn’t explicitly stated. Now, the question is, what’s next after AP Calculus BC? If you are mainly interested in the humanities or want to go into research sciences, then AP Statistics is a great option. Congratulations, you’re done with this unit, and as a result, you’ve also reached the end of AP Calculus BC! Now, you have all the tools you need to ace that AP test this May! ![]()
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