11/23/2023 0 Comments Calc bc sequences and series![]() ![]() Let's say that we want to look at a, we want to figure out Now, with that out of the way for review, let's tackle something a That, even though this is an infinite sum, it willĬonverge to a finite value. On and on and on and on until it makes sense ![]() To go down by that common, or it's going to be multipliedīy that common ratio, and it's going to decrease Of R is less than one then each term here is going Logical sense that, look, if that absolute value ![]() We've proven it as well, but it also makes And if the absolute value of R is greater than or equal to one, then the series diverges. If the absolute value of the common ratio is less than one, then the series converges. And what's interestingĪbout this is we've proven to ourselves, in the videosĪbout geometric series, that if the common ratio, If it is not review, I encourage you to watch the videos on geometric series. When you go from one term to another you are just multiplying, youĪre just multiplying by R, and this is all review. To R, or R to the first power, and you see that here. That same thing with the N, that's just going to be equal Well, this is just going to be equal to R. Which is the ratio between consecutive terms, is going There are a few things we'veĪlready thought about here. Plus R to the K plus one plus R to the K plus two and keep going on and on and on forever. Infinity of R to the N, which would be R to the K Infinite geometric series starting at N equals K to We already have a lot of experience with the geometric series. Let me know if that didn't fully help and I can try explaining differently If you'd like to see it on a smaller scale, try (100x^2 + 100x)/(x^3) it starts with the numerator being larger, but eventually the denominator is bigger. if the denominator keeps getting bigger than the numerator than eventually it will equal 0. What does that mean? well, a large denominator makes the fraction get closer and closer to 0. the numerator may start out bigger, but as you head toward infinity, the larger exponent will always make a bigger term, so the denominator will get bigger than the numerator. Nowwhat happens as n gets bigger and bigger? n^11 will always be bigger than an^9 + bn^8 +. the n^10 in the numerator and denominator kinda cancel out, or at least they will be the same number no matter what n is, so we only need to worry about the rest. You could maybe look at it as n^10 + n^11. + z where a through z are some real numbers. Let h be a function for which all derivatives exist at x = 1.So, the numerator is a massive polynomial, but the largest term is n^10, so it will be n^10 + an^9 +bn^8 +. 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. 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. 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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