MAC 2282 USF Determine the Interval of Convergence Calculus Questions

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Mathematics

MAC 2282

University of South Florida

MAC

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CALC. 2. QUIZ 5. SPRING 2021 MUST SHOW ALL DETAILS AND SIMPLIFY ALL ANSWERS. NO WORK, NO CREDIT. 1. Interval of convergence, including the endpoints (if any) for the power series: ∑∞ 𝑛=0 𝑛+1 3𝑛 (𝑥 + 1)𝑛 . 2. Compute the following integral with 6 decimal places: 0.5 𝑥 3 ∫0 1+ 𝑥 5 𝑑𝑥 . 3 3. Consider the binomial series associated with the function √1 + 𝑥 . a) Write the series in summation notation, simplifying the coefficients AS MUCH AS POSSIBLE; b) Check directly the radius of convergence; c) Use 3 the series to compute √1.3 with 3 decimal places; justify . THE FOLLOWING PROBLEM IS OPTIONAL, FOR 10 EXTRA CREDIT POINTS. MUST JUSTIFY IN COMPLETE DETAIL. 4. Consider the binomial series associated to the function ( 1 + 𝑥 )𝑟 (r a fixed real number.) The sum in the interval ( – 1 , 1 ) is some function g(x). Verify that 𝑔(𝑥) = ( 1 + 𝑥 )𝑟 . Suggestions: a) Show that ( 1 + 𝑥 )𝑔′ (𝑥) = 𝑟 𝑔(𝑥) b) Show that the derivative of the function 𝑔(𝑥)/( 1 + 𝑥 )𝑟 is identically zero in the open interval ( - 1 , 1 ). NOTE: The idea is similar to one of the arguments we gave for the exponential. Main difficulty is (a), which you have to check VERY VERY CAREFULLY. MAC-2282. TEST 4. SPRING 2021 NAME:____________________________________________________ MUST SHOW ALL WORK IN DETAIL: NO WORK, NO CREDIT. 1. Determine the interval of convergence, including the endpoints, for: ∑∞ 𝑛=0 𝑛2 (𝑥 +2)𝑛 4 𝑛 √2𝑛+1 . 2. Find the first THREE non-zero terms of the MacLaurin series of the function 𝑓(𝑥) = 𝑒− 2 𝑥 1− 𝑥 + 𝑥 2 . 3. Determine the first 4 non zero terms of the MacLaurin series of: 𝑓(𝑥) = √4 − 𝑥 2 . 4. Use power series to compute the limit: lim𝑥− >0 12 ln(1+𝑥)−12 𝑥+6 𝑥 2 −4 𝑥 3 3 𝑥4 .MUST JUSTIFY IN DETAIL. 0.5 5. Compute the following integral with 4 decimal places: ∫0 cos( 𝑥 5 ) 𝑑𝑥 . MUST JUSTIFY IN DETAIL. 6. For the function 𝑓(𝑥) = ln ( 1 + 𝑥) : a) Find the partial sum up to third power of the Taylor series of f around the point a = 1 ; b) Using the Taylor remainder 𝑅3 (𝑥) , estimate the error, when using this partial sum 𝑆3 (𝑥) , on the whole interval [ 0.5 , 1.5 ]. NAME PARIS APCHAT 2 h Me Cautest seits i given by f(x) = fro) + "(0) "(0), 21 3! t" (0) n! #70) agosto e 1-2 Fist fennu o Melausen series, - 200) fro) e
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