Math 255 Calculus II Fall 2024 Third Test Topics PDF

Math 255 – Calculus II Topics for the Third (Non-Gateway) Test Chapter 5: Infinite Series, Sections 5.2 - 5.6 ˆ You should know how to calculate n! (n-factorial), i.e. n! = n ∗ (n − 1) ∗ (n − 2) ∗ · · · ∗ 3 ∗ 2 ∗ 1 for n ≥ 1 and 0! = 1 by definition. ˆ You...

Math 255 – Calculus II Topics for the Third (Non-Gateway) Test Chapter 5: Infinite Series, Sections 5.2 - 5.6 ˆ You should know how to calculate n! (n-factorial), i.e. n! = n ∗ (n − 1) ∗ (n − 2) ∗ · · · ∗ 3 ∗ 2 ∗ 1 for n ≥ 1 and 0! = 1 by definition. ˆ You should be able to use l’Hopital’s Rule to evaluate a limit, when needed. ˆ When evaluating a limit of a rational expression where the variable goes to infinity and the numerator and denominator are polynomials, you should be familiar with and able to use the “top heavy”, “bottom heavy”, or “same” rules to determine the value of the limit. ˆ You should be able to explain what a series is, and the difference between a sequence and a series. ∞ ˆ You should know what a partial sum sn is with respect to the series X an and why it is important to n=1 determining if the series converges. ˆ You should know what it means for a series to converge or diverge, and how this is different than the convergence of a sequence. ˆ You should be able to expand (i.e. write out the first few terms of a series) as well as determine the pattern term an of a given series. ∞ ˆ You should be able to identify a geometric series, i.e. a series of the form X arn and under what conditions n=0 a it converges or diverges. (Recall, it converges to if |r| < 1, else it diverges.) 1−r ˆ Given a geometric series starting with an index value other than n = 0 you should be able to write out the ∞ X first few terms to determine the values of a and r in order to rewrite it as arn. n=0 ˆ You should know what constitutes a telescoping series and how to find sn , the nth partial sum, and thus find whether the series converges or diverges. ˆ You must be able to use partial fraction decomposition to break a fraction term, an an , into the difference of two fractions that lead to a telescoping series. ˆ You should know how to use the nth Term Test for Divergence. ˆ You should understand that the nth Term Test for Divergence can NOT be used to conclude that a series converges. ˆ You should know how to use the Integral test to determine convergence or divergence of a series, and when it is a good choice of test to use. ∞ 1 ˆ You should be able to recognize a p-Series, namely a series of the form X and know under what n=1 np conditions a p-series converges or diverges. ˆ You should know what we mean by the harmonic series and whether it converges or diverges. ˆ You need to be able to use the Limit Comparison Test to determine convergence or divergence of a series, and when it is a good choice of test to use. ˆ You should recognize when and how to use the Alternating Series Test to determine convergence or divergence of a series.  ˆ You should know how to use the ratio test and under what conditions it indicates a series converges or diverges. You should also know when the ratio test is inconclusive. ˆ You should recognize attributes of a series that make it likely that the ratio test will be helpful in determining it’s behavior. ˆ You should be able to reduce/simplify ratios of factorials using the definition of n!. ˆ You should be able to understand the very handy table of tests for series convergence towards the end of Sec- tion 5.6 of the text (https://openstax.org/books/calculus-volume-2/pages/5-6-ratio-and-root-tests). ˆ You should know which series we can actually calculate (i.e. for which series we can calculate the value of the infinite sum), and which series tests only tell us the behavior of the series (convergent or divergent), but not what they would converge to. Chapter 16: Power Series ∞ ˆ You should know what we mean by a power series and its general form X cn (x − a)n. n=0 ˆ You should know what we mean by the center of a power series, a, and how to determine the center of a given power series. ˆ You should be able to give examples of power series. ˆ You should be able to explain what it means for a power series to converge; in particular, you need to be explain what we mean by the radius of convergence of a power series. ˆ Given a power series, you need to be able to determine its radius of convergence. ˆ You need to be able to use the ratio test to determine the radius of convergence of a power series. (This includes knowing the relationship between the value of the limit from the ratio test and the radius of convergence, and why that is the case. i.e. if L = 0 then R = , and if L = ∞, then R = , and if L = |x − a| ∗ C for some constant C, then know how to find R. ˆ You should know the behavior of some well-known series, particularly p-series, the harmonic series and the alternating harmonic series. ∞ f (n) (c) ˆ You should know the formula for the Taylor Series for f (x) centered at c, i.e. X (x − c)n , and n=0 n! be able to use it to derive a Taylor Series for a specified function. This includes being able to explain what the f (n) (c) term represents with respect to the function f (x). ˆ You should be able to explain why a Taylor Series is a special case of a power series. ˆ You should know what we mean by a Maclaurin series and how this is a special case of a Taylor Series. ˆ You should know and understand that if a Taylor (or Maclaurin) series for f (x) converges for some interval of convergence, then the Taylor (or Maclaurin) series value IS EQUAL TO the value of f (x) for an x in that interval. ˆ Given a Taylor (or Maclaurin) series for some function f (x), you should be able to identify where the series was centered, i.e. c, and the values of f (c), f ′ (c), f ′′ (c), etc. ˆ You should be able to use the Maclaurin series for ex , sin(x), cos(x), and 1−x 1 as well as substitution, addition and subtraction, multiplication and division, and differentiation or integration to find Maclaurin series for similar or related functions without having to use the standard process (which involves taking a lot of derivatives to start... ). ˆ You need to be able to explain what we mean by the nth order Taylor (or Maclaurin) polynomial (i.e. stopping our series after the nth degree term). ˆ You need to be able to approximate the error the value of an alternating Taylor (or Maclaurin) series evaluated at a specific input versus the value of the nth order Taylor (or Maclaurin) polynomial at the same input. Note this is related to the value of the “first dropped term”. ˆ You need to be able to give reasons why it’s handy to know and be able to use Taylor (or Maclaurin) polynomials in applied problems.