NAME____________________________________ SCHOOL__________________________________ TEST 3 Monday, April 9, 2007 Fax the test answers to: Fax 515-294-5454 and also Snail Mail your test answers to: Irvin Roy Hentzel Department of Mathematics 432 Carver Hall Iowa State University Ames, Iowa 50011-2064 1. (a) Write the power series for each of these. (b) Give the radius of convergence. (c) Give an error bound for truncating the series at degree 4 on the interval ( -1/3, 1/3 ). 1 (i) -------- 1-x 2 3 n (a) 1 + x + x + x + ... + x + ... (b) R = 1 (5) f (c) 5 (c) Error <= --------- (1/3) 5! f(x) = (1-x)^(-1) f'(x) = (1-x)^(-2) f''(x) = 2(1-x)^(-3) f'''(x) = 3!(1-x)^(-4) f''''(x) = 4!(1-x)^(-5) f'''''(x) = 5!(1-x)^(-6) 5! -------- 6 (2/3) 5 Error <= ------------ (1/3) = 3/64 = 0.046875 5! (ii) ln(1+x) 1 2 3 4 n n ---- = 1 - x + x - x + x ... +(-1) x .... 1+x 2 3 4 n x x x n+1 x (a) x - ---- + --- - --- ..... (-1) ------ .... 2 3 4 n (b) R = 1 5 (1/3) (c) error < --------- = 1/1215 = 0.000823045 5 (iii) ArcTan[x] 2 4 6 n 2n (a) ArcTan[x] = INT ( 1 - x + x - x .... (-1) x ... 3 5 7 9 2n+1 x x x x n x x - ---- + ---- - ---- + --- ..... (-1) ----- ... 3 5 7 9 2n+1 (b) R = 1 5 (1/3) (c) Error < ------ = 1/1215 = 0.000823045 5 x (iv) e 2 3 4 n (a) 1 + x + x /2 + x /3! + x /4! + .... + x /n! + ..... (b) R = Infinity 1/3 1/3 e 5 E (c) Error <= ------ (1/3) = ------ = 0.0000478605 5| 29160 (v) Sin[x] 3 5 2n+1 (a) x - x /3! + x /5! - .... (-1)^(n+1) x /(2n+1)! (b) R = Infinity 5 (1/3) (c) Error <= ------- = 1/29160 = 0.0000342936 5! (vi) Cos[x] 2 4 n 2n (a) 1 - x /2! + x /4! - .... (-1) x /(2n)! (b) R = Infinity 6 (1/3) -6 (c) Error <= ------- = 1/524880 = 1.9052 * 10 6! (viii) Sinh[x] 3 5 2n+1 (a) x + x /3! + x /5! + .... x /(2n+1)! (b) R = Infinity 1 Cosh[-] Cosh[1/3] 5 3 (c) Error <= --------- (1/3) = ------------- = 0.0000362165 5! 29160 (viii) Cosh[x] 2 4 2n (a) 1 + x /2! + x /4! + .... x /(2n)! (b) R = Infinity 1 Sinh[---] Sinh[1/3] 5 3 (c) Error <= ---------- (1/3) = ---------- = 0.0000116441 5! 29160 1/2 1/2 2. (a) Write the power series for y = x Sin[ x ]. (b) What is its radius of convergence. (c) What is the error by truncating the series at degree 4 on the interval [-1/3,1/3]; 3/2 5/2 7/2 (2n+1)/2 1/2 x x x n x (a) Sqrt[x] ( x - --- + ----- - ------ .... (-1) ----------- ... 3! 5! 7! (2n+1)! 2 3 4 n+1 x x x n x x - ----- + ----- - ----- ..... (-1) ------ + .... 3! 5! 7! (2n+1)! (b) R = Infinity 5 (1/3) -8 (c) error <= ----- = 1/88179840 = 1.13405 10 9! 2 3. (a) Write the power series for y = x ln( 1-x ). (b) What is its radius of convergence. 1 2 3 n ------- = 1 + x + x + x + .... + x + .... 1-x 2 3 4 n+1 -log(1-x) = x + x /2 + x /3 + x /4 + ... x / (n+1) + .... 2 2 4 6 8 2n+2 log(1-x ) = -x - x /2 - x /3 - x /4 - ... -x / (n+1) 2 3 5 7 9 2n+3 x log(1-x ) = -x - x /2 - x /3 - x /4 - ... - x / ( n+1) (b) R = 1 2 (x ) 4. (a) Write the degree 3 MacLaurin polynomial for y = e (b) Give a bound on the accuracy of your approximation on (-1/2,1/2). 2 ( x ) f(x) = e 1 2 (x ) f'(x) = e 2x 0 2 _ _ (x ) | 2 | f''(x) = e | 4x + 2 | 2 |_ _| 2 _ _ (x ) | 3 | f'''(x) = e | 8 x + 12 x | 0 |_ _| _ 2 | (x ) | 4 2 2 f''''(x) = e | ( 16 x + 24 x + 24 x + 12 ) |_ 2 2 x 2 (a) 1 + ----- = 1 + x 2 1/4 1/4 (b) Error <= e ( 16/16 + 48/4 + 12) 4 25 e -------------------------- (1/2) = ------- = 0.0835954 4! 384 5. Find the radius of convergence of n Infinity 3 2n SUM ------ x n=1 n 1+2 a n+1 2n+2 n n n+1 3 x 1 + 2 2 1 + 2 2 ----- = ---------- * -------- = x 3 ----------- ----> x 3/2 a n+1 n 2n n+1 n 1 + 2 3 x 1 + 2 R = Sqrt[2/3] 6. Prove that the series converges n Infinity Cos[ 2 ] SUM ------------ n=1 2 n n Infinity | Cos[ 2 ] | Infinity 1 SUM |------------ | <= SUM ----- n=1 | 2 | n=1 2 | n | n This is a p-series, p=2 and it converges. The original series is absolutely convergent by the comparison test. Therefore the original series converges. 7. Prove that the series diverges Infinity 2n+1 SUM -------- n=1 2 n Infinity 2n+1 Infinity 2 Infinity 1 SUM -------- = SUM ----- + SUM ----- n=1 2 n=1 n n=1 2 n n The first sum on the right hand side is the harmonic series which diverges. The second sum is a p-series, p=2 which converges. Therefore the original series diverges. th 8. State the n term test for divergence. Infinity If SUM a converges, then Limit a = 0 n=0 n n->Infinity n th 9. State the n term test for convergence of an alternating series. Infinity n If a decreases to zero, then SUM (-1) a converges. n n=0 n 10. State the integral test. If f is a decreasing positive function, then Infinity Infinity INT f(x) dx exists <=====> SUM f(n) exists x=0 n=0 11. State the ratio test. a n+1 If a > 0 and Limit ------ = r then n n->Infinity a n Infinity if r < 1, SUM a converges. n=1 n Infinity if r > 1, SUM a diverges. n=1 n if r = 1, the series may diverge or may converge.