11/15/2023 0 Comments Calc bc sequences and series![]() 8.8: Taylor Series The difference between a Taylor polynomial and a Taylor series is the former is a polynomial, containing only a finite number of terms, whereas the latter is a series, a summation of an infinite set of terms.8.7: Taylor Polynomials A Taylor polynomial is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.We start this new approach to series with a definition. Given a value of x, we evaluate f(x) by finding the sum of a particular series that depends on x (assuming the series converges). 8.6: Power Series So far, our study of series has examined the question of "Is the sum of these infinite terms finite?,'' i.e., "Does the series converge?'' We now approach series from a different perspective: as a function.We start with a very specific form of series, where the terms of the summation alternate between being positive and negative. 8.5: Alternating Series and Absolute Convergence In this section we explore series whose summation includes negative terms.This section introduces the Ratio and Root Tests, which determine convergence by analyzing the terms of a series to see if they approach 0 "fast enough.'' 8.4: Ratio and Root Tests The comparison tests of the previous section determine convergence by comparing terms of a series to terms of another series whose convergence is known.8.3: Integral and Comparison Tests There are many important series whose convergence cannot be determined by these theorems, though, so we introduce a set of tests that allow us to handle a broad range of series including the Integral and Comparison Tests.Most series that we encounter are not one of these types, but we are still interested in knowing whether or not they converge. 8.2: Infinite Series This section introduces us to series and defined a few special types of series whose convergence properties are well known: we know when a p-series or a geometric series converges or diverges.In mathematics, we use the word sequence to refer to an ordered set of numbers, i.e., a set of numbers that "occur one after the other.'' For instance, the numbers 2, 4, 6, 8. 8.1: Sequences We commonly refer to a set of events that occur one after the other as a sequence of events.Let h be a function for which all derivatives exist at x = 1.\) The third-order Taylor polynomial P 3 ( x) for sin x about isĤ8. ![]() (C) If the terms of an alternating series decrease, then the series converges.Ĥ7. (B) If a series is truncated after the nth term, then the error is less than the first term omitted. (A) If converges, then so does the series Which of the following statements is true? Which of the following alternating series diverges?Ģ6. For which of the following series does the Ratio Test fail?Ģ5. Which of the following series diverges?Ģ4. Which of the following series converges?Ģ3. (E) Rearranging the terms of a positive convergent series will not affect its convergence or its sum.Ģ2. (D) If 1000 terms are added to a convergent series, the new series also converges. (C) If and converge, so does where c ≠ 0. Course Information Sequences Series and Convergence The Integral Test and p-series Comparisons of Series Alternating Series The Ratio and Root Tests. Which of the following statements about series is false? Directions: Some of the following questions require the use of a graphing calculator.Ģ1. Replace the first sentence in Question 19 by “Let f be the Taylor polynomial P 9 ( x) of order 9 for tan −1 x about x = 0.” Which choice given in Question 19 is now the correct one? Then it follows that, if −0.5 tan −1 x if x 0Ģ0. Let f be the Taylor polynomial P 7 ( x) of order 7 for tan −1 x about x = 0. If the series tan −1 is used to approximate with an error less than 0.001, then the smallest number of terms needed isġ9. If an appropriate series is used to evaluate then, correct to three decimal places, the definite integral equalsġ8. The coefficient of x 4 in the Maclaurin series for f ( x) = e − x /2 isġ7. ![]() Which of the following expansions is impossible?ġ6. Chapter 9 Infinite Sequences and Series BC 9.1 Sequences and Infinite Series 9.2 The Integral Test and p - Series 9.3 The Comparison Tests 9.4 Alternating. Which of the following series diverges?ġ3. Which of the following series diverges?ġ1. Which of the following statements about series is true?ġ0. is a series of constants for which Which of the following statements is always true?ĩ. Which of the following sequences diverges?ĥ. Review of sequences will enhance understanding of series.Ĥ. ![]() We have nevertheless chosen to include the topic in Questions 1–5 because a series and its convergence are defined in terms of sequences. ![]() Note: No questions on sequences will appear on the BC examination. Directions: Answer these questions without using your calculator. Calculus AB and Calculus BC CHAPTER 10 Sequences and Series ![]()
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