Math 270 Name __________________________ Lab #4 This week, we continue our exploration of series solutions by adding two distinct types of series to our repertoire of available mathematical processes with the Taylor and Fourier Series . The Taylor and Fourier Series are, arguably, among the most important ideas in mathematics that you will encounter during this class as they have great practical use in band pass filters, spectrum analysis, video encoding, and MP3 and MP4 compression techniques. Be prepared to use them in several of your upcoming electronics courses. In this lab, we address concepts of Operations and Computations with (review) Series, Maclaurin Expansion, and (new) Taylor, and Fourier Series. Remember to SHOW ALL YOUR WORK TO EARN FULL CREDIT, because providing only the answers will earn approximately 1/3 of the available credit. Table 1 Brief Table of Maclaurin and Binomial Series Expansions Function: Series Expansion: Note: n! = n(n-1)(n-2)(n-3) ... (3)(2)(1), for example, 4! = 4*3*2*1=24 u 1 u 2 u3 u 4 u n+1 1+ + + + + ...+ +... u 1 ! 2! 3! 4 ! ( n+1 ) ! e 2 n+1 u 1 u3 u 5 n u + +...+ (1 ) +... sin ( u ) 1! 3! 5! ( 2 n+1 ) ! 2n u2 u 4 u6 n (u ) 1 + + ...+ (1 ) +... cos ( u ) 2! 4 ! 6! ( 2n ) ! (1 )n ( un +1 ) u2 u3 u 4 u + + ...+ +...|u|<1 ln ( 1+u ) 2! 3! 4! n+1 ! n n1 (n2) 2 an u0 + a u1 u +...+n 1 un1 +a 0 un 1! a+u review: series operations and maclaurin expansions. 3 8 (n +1) 1. find the sum of geometric 5 7 () page using appropriate formula: math 270 name __________________________ 2. for lab #4 : (a) calculate show first six partial sums (remember, these are sn terms, not terms), (b) determine if it converges or diverges, (c) state your mathematical reasoning convergence divergence. s0 =_______________________ s1 =_______________________ s2 =_______________________ s3 =_______________________ s4 =_______________________ s5 =_______________________ converge diverge? reasoning: ___________________________________________ 3. one expansions above, write three terms sin x3) 4. each values below utilizing four then compare this value with calculator value. note: points based upon calculated estimated values, which different from exact values. function expansion e.35 terms: cos 3.57 3.54) value: (for comparison purposes only) week 4 material 5. taylor approximation f (x) =cos x) , x with, 6. non-zero e 7. at 3, utilize to approximate 10 9>