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Unformatted text preview: MATH1ZA3 Test #3 Guide Jump Start Academic Solutions Contents I Math 1ZA3 Test #3 Guide 3 Test Break Down 4 Test Strategy 5 II Infinite Series 9 Study Notes 10 Problems 13 III Infinite Sequences 17 Study Notes 18 Problems 20 IV Power Series 24 Study Notes 25 Problems 26 V Riemann Sums 33 Study Notes 34 1 CONTENTS 2 Problems 37 VI Integration by Substitution 45 Study Notes 46 Problems 47 VII Exponential Growth and Decay 51 Study Notes 52 Problems 53 Part I Math 1ZA3 Test #3 Guide 3 Test Break Down Mark Percentage of Sample Tests Relative Difficulty: Straight Forward , Medium , Tricky 1. 29%  Power Series (sums with s in them) 12%  Find interval/radius of convergence 6%  Power Series Representation 6%  Find sum of power series 5%  Other 2. 27%  Infinite Series 17%  Converge / Diverge 6%  Find sum of infinite series 4%  Other 3. 24%  Integration/Antidifferentiation 12%  Given a or find 4%  Given a graph of find graph of F 4%  Change limit of sums to a definite integral 4%  Indefinite Integration 4. 8%  Infinite Sequences 4%  Converge/Diverge 4 %  Recursively Defined Sequences 5. 4%  Exponential Growth/Decay 6. 4%  Induction 7. 2%  Sigma Notation 8. 2%  Differential Equations 4 Test Strategies Plan of Attack Try and do the questions on the test in this order: 1. Integration/Antidifferentiation Questions 2. Infinite Sequences 3. Exponential Growth/Decay 4. Differential Equations 5. Power Series 6. Infinite Series Find Interval/Radius of Convergence To find the radius of convergence, generally proceed as follows: 1. Determine 2. Use ratio test 3. Set the result of ratio test to be less than 1 Power Series Representation To find the power series representation, generally proceed as follows: 1. Rewrite the question (if needed) into a form that looks like a known function (for example, 1 1 ) 2. If you know the representation of a function appearing in the question, use it 3. Manipulate formula (multiply by , etc) to match formula 5 6 Sum of Power Series To find the sum of a power series, generally proceed as follows: 1. Try and rewrite (by integrating/differentiating, etc) the question to make it look like =0 2. Use the fact that, =0 = 1 1 Convergent / Divergent Series To test a series for convergence/divergence try to use the following tests in this order: 1. Divergence Test 2. (If series is alternating) Alternating Series Test 3. Integral Test 4. Ratio Test 5. Root Test 6. Comparison Test 7. Limit Comparison Test Sum of Infinite Series If you are asked to find the sum of an infinite series, the only three ways you can do that is: 1. Write it as a telescoping sum 2. Write is a a geometric series and use the fact that, =1 1 = 1 3. If they give you the partial sum , take lim 7 Estimating Sums If you are asked to find the number of terms to estimate a sum, the only three ways of doing it are: 1. If its an alternating series, use the Estimate for Alternating Series Theorem1....
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 Spring '11
 CHILDS
 Math, Power Series

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