Math 166 SI review With Rosalie 

Section 8.6 – Trapezoid Rule Used to estimate the area under a curve 𝑇= ∆𝑥 2 ( 𝑦 0 + 2𝑦 1 + 2𝑦 2 + … + 2𝑦 𝑛−1 + 𝑦 𝑛 ) ∆𝑥= 𝑏−𝑎 𝑛

Example: Estimate the area under the curve of 𝑦= 𝑥 2 from 𝑥=1 to 𝑥=2 using the trapezoid rule with n = 4.

8.7 Improper Integrals Improper integrals are VERY important There are two cases for improper integrals When an integral has infinite limits of integration 0 ∞ 𝑥 𝑑𝑥 When an integral has points within the limits of integration that it can’t exist −1 1 1 𝑥 2 𝑑𝑥 These integrals are solved using limits to replace the “issue” values in the limits of integration Sometimes the intergrals have to broken into 2 integrals in order to solve.

Example Solve the integral: 1 ∞ ln 𝑥 𝑥 2 𝑑𝑥

10.1 Sequences A sequence is a series of numbers, usually following a pattern 2,4,6,8,10,12,14,16,18 … 2n 1, −1 2 , 1 3 , −1 4 , 1 5 , −1 6 … (−1) 𝑛+1 1 𝑛 You can test a series for convergence by taking a limit Other sequence rules: Sandwich Theorem: if a sequence above and below a third sequence both converge to a number, then the third sequence converges to the same number Monotonic Sequence Theorem: If a sequence is both bounded and monotonic, then the sequence converges

Example Which of these sequences converge and which diverge? A) 𝑎 𝑛 =1 B) 𝑎 𝑛 = 1 𝑛 C) 𝑎 𝑛 =𝑛

10.2 - Series A series is the all of the terms of a sequence added together 𝑎 𝑛 A partial sum is a certain number of terms added together Ie, the fifth partial sum is the first 5 terms added together A geometric series is very important and it follows : 𝑎𝑟 𝑛 Where a and r are constants and n varies. A geometric series as long as |r| is less than 1

10.2 - Series A telescoping series works out so that the inner terms cancel by subtraction The 𝑛 𝑡ℎ term test is incredibly important You run this test by taking the limit: lim 𝑛→∞ 𝑎 𝑛 If the limit is not 0, the series diverges If the limit is 0, the series may converge or diverge A p-series is also very important. It follows: 1 𝑛 𝑝 and converges when p > 1, diverges when p ≤ 1

Example Test for convergence: A) 𝑛=1 ∞ 1 B) 𝑛=1 ∞ 5 ( 1 2 ) 𝑛 C) 𝑛=1 ∞ 1 𝑛(𝑛+1)  Hint: Use partial fractions

10.3 – The Integral Test The integral test is a very useful test It only works if the series is: Positive Decreasing (or non-increasing) Continuous For this test, you take the integral over its domain and if the integral converges to a number, the series converges. If the integral diverges, so does the series NOTE: THE SERIES DOES NOT CONVERGE TO THE SAME NUMBER AS THE INTEGRAL

Example Use the integral test to determine whether the series converges or diverges 𝑛=1 ∞ 1 𝑛 2

10.4 – Comparison Tests The Comparison test is a lot like the monotonic sequence theorem If 𝑎 𝑛 > 𝑏 𝑛 𝑎𝑛𝑑 𝑎 𝑛 𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑠, 𝑡ℎ𝑒𝑛 𝑠𝑜 𝑑𝑜𝑒𝑠 𝑏 𝑛 If 𝑏 𝑛 > 𝑎 𝑛 𝑎𝑛𝑑 𝑎 𝑛 𝑑𝑖𝑣𝑒𝑟𝑔𝑒𝑠, 𝑡ℎ𝑒𝑛 𝑠𝑜 𝑑𝑜𝑒𝑠 𝑏 𝑛 There are 2 tricks to the limit comparison test: 1) Picking the right series to compare with 2) Proving that the series is greater or less than the series you’re dealing with

10.4 – Comparison Tests The limit comparison test involves, you guessed it, limits. For this test you must also pick a similar series. If your series is 𝑎 𝑛 , then you choose 𝑏 𝑛 . Next: lim 𝑛→ ∞ 𝑎 𝑛 𝑏 𝑛 = L If 0 < L < ∞ then 𝑎 𝑛 and 𝑏 𝑛 either both converge or both diverge If L = 0 and 𝑏 𝑛 converges, then so does 𝑎 𝑛 If L = ∞ and 𝑏 𝑛 diverges, then so does 𝑎 𝑛

Examples A) Use the comparison test to determine whether the series converges or diverges: 𝑛=1 ∞ 1 𝑛−1 B) Use the limit comparison test to determine whether the series converges or diverges 𝑛=1 ∞ 1 2 𝑛 −1

10.5 - The ratio test The ratio test works by basically making a comparison with a geometric series. The ratio test says that: lim 𝑛→∞ 𝑎 𝑛+1 𝑎 𝑛 = 𝜌 If 𝜌 < 1, then the series 𝑎 𝑛 converges If 𝜌 > 1, then the series 𝑎 𝑛 diverges If 𝜌 = 1, then the test is inconclusive and you need to use a different test Ratio test is very useful when working with factorials. 10.5 also covers the root test, but you won’t be responsible for it. It can be a helpful test (and it’s pretty easy) so you might want to take a few minutes and learn it on your own.

Example Analyze the series using the ratio test: 𝑛=1 ∞ 𝑛!𝑛! 4 𝑛 2𝑛 !

10.6 – Alternating Series For an alternating series, you can only use alternating series test or nth term test First, you should test for absolute convergence. You do this by taking the absolute value of the series, and testing for convergence with any of the tests you know. If a series converges absolutely (meaning if the summation of all terms if they are all positive), then it converges no matter what. If the series does not converge absolutely, then you need to check for conditional convergence (meaning it only converges if some of the terms are negative) using the alternating series test which says: The ser𝑖𝑒𝑠 (−1) 𝑛 𝑢 𝑛 converges if: All of 𝑢 𝑛 is positive The sequence 𝑢 𝑛 is eventually decreasing 𝑢 𝑛  0

Example Determine whether the following series converges absolutely, conditionally, or not at all. 𝑛=1 ∞ (−1) 𝑛 1 𝑛

10.7 – Power Series A power series is simply a series with an extra variable, and the goal is to determine for what values the series will converge Usually uses ratio test

Example Determine the radius of convergence for : 𝑛=1 ∞ − 1 2 𝑛 (𝑥−2) 𝑛

10.8 - Taylor and Maclaurin series A Taylor series is represented by: 𝑘=0 ∞ 𝑓 𝑘 (𝑎) 𝑘! (𝑥−𝑎) 𝑘 Where a is some constant number and f(a) is some function A Maclaurin series is simply a Taylor series with a=0

Example Find the Taylor series generated by 𝑓 𝑎 = 𝑒 2𝑎 at a=0 and at a=2