Review Test 3: 6 Multiple Choice: Series: Convergence, Divergence, Absolute Convergence, Conditional Convergence, Sum (geometric, telescoping) Free Response: 1. L’Hopital – recognize and apply 2. Improper Integrals – recognize type of improper integral, compute improper integrals using CORRECT notation 3. Series – use known tests (alternating series, root, ratio, p-series, limit comparison, integral, basic comparison, geometric, basic divergence) to determine convergence 4. Taylor Polynomials and Series – Give Taylor polynomials using given information (values, functions, etc); be able to find the error; radius and interval of convergence Find a Taylor series for cos (x2) centered at x = 0: Find a Taylor series for e2x centered at x = 0: 2 Find the Taylor polynomial P5(x) for f (x) = xcosx . –x Find the nth Taylor polynomial Pn for the function f (x) = e Find the nth Taylor polynomial Pn for the function f (x) = sinh x Find the nth Taylor polynomial Pn for the function f (x) = ln (1 – x) Give the 5th degree Taylor polynomial for f (x) = sin(x) centered at 0. Give the 5th degree Taylor polynomial for f (x) = ex centered at 0. Give the 5th degree Taylor polynomial for f (x) = ln(x+1) centered at 0. Give the 5th degree Taylor polynomial for f (x) = cos(x) centered at 0. f,f',f''(212221) =−( ) =( ) =− Give the 2nd degree Taylor polynomial for f centered at 2. Rewrite f (x) = 3x3 +2x2 – x + 1 in powers of (x – 2). Create the 3rd degree Taylor Polynomial for f (x) = arctan(x) centered at x = 0. f n1+ c ( ) n1+ Rxn ( ) = x (n1+ )! Use the Lagrange formula to find the smallest value of n so that the Taylor polynomial of degree n for f (x) = cos (x) centered at x = 0 can be used to approximate f (x) within 10 –4 at x = 1. Use the Lagrange formula to find the smallest value of n so that the nth degree Taylor Polynomial for f (x) = ln (1 + x) centered at x = 0 approximates ln (2) with an error of no more than 0.01. n1+ ∞ −1 Which term is truncated if we want to approximate the sum of ( ) ∑ 3 n1= 2n− 1 1 with an error of less than ? 1000 1. State the indeterminate form and compute the following limits : ln( n4+ ) a. lim n→∞ n2+ 2 b. lim 3n n n→∞ ( ) 2n ⎛⎞3 c. lim⎜⎟1 + n→∞ ⎝⎠n x2x− sin( ) d. lim x0→ x2x+ sin( ) 2 e1x − e. lim x0→ 2x2 x ⎛⎞1 f. lim + ⎜⎟ x0→ ⎝⎠x 3ex3/ −( 3+ x ) g. lim x0→ x 2 x 2 h. lim x→∞ lnx 1xe+−x i. lim x0→ xe( x − 1) arctan( 4x ) j. lim x0→ x ∞ 1 2. Give the exact value of . ∑ n n=0 2 ∞ 1 3. Give the exact value of ∑ . n=2 nn( +1) ∞ cos πn 4. Give the exact value of ( ). ∑ n n=2 3 5. Evaluate each improper integral, and explain why it is improper. Use correct notation. 2 1 a. dx ∫−1 x 2 1 1 b. dx ∫0 6 1x− 7 14 c. dx ∫5 2 ( x6− ) 27 d. xdx−23/ ∫ 0 4 1 dx e. ∫ 0 4x− ∞ 1 f. dx = ∫0 1x+ 2 5 dx g. = ∫2 x2− Notes for series “growth”: Let p(k) be a polynomial in k. rk for r > 1 grows much faster than p(k) k! grows much faster than rk, p(k) kk grows much faster than the others Hence, pk( ) pk( ) pk( ) ,, ∑∑∑rkkkk ! rrkkk! ,, ∑∑∑k ! kkkk ALL converge rapidly. Determine if the following series converge absolutely, converge conditionally, or diverge? n+1 ∞ −1 n a. ( ) ∑ n=1 n + 3 ∞ cosπ n b. 2 ∑ n=1 n n ∞ 4n( − 1) c. ∑ 2 n0= 3n++ 2n 1 n ∞ 31− d. ( ) ∑ 2 n0= 3n++ 2n 1 n ∞ 3n− 1 e. ( ) ∑ 2 n0= 3n++ 2n 1 n ∞ ⎛⎞n ⎛⎞n f. ∑⎜⎟41( − ) ⎜⎟ n0⎜⎟n3+ = ⎝⎠⎝⎠ n ∞ ⎛⎞21( − ) arctan n g. ⎜⎟ ∑ 23 n0= ⎜⎟3n++ n ⎝⎠ n n ∞ ⎛⎞( −13) h. ⎜⎟ ∑ n n0= ⎜⎟43n+ ⎝⎠ n ∞ ⎛⎞( −13) i. ∑⎜⎟ n0= ⎜⎟n2++ln n2 ⎝⎠( ) ( ) n ∞ (−1n) ! j. ∑ n2= (n1+ )! n ∞ (−1) k. ∑ n2= 3n+ 2 n ∞ −110n2 l. ( ) ∑ n n2= 3 n ∞ (−13) n m. ∑ n2= n! n ∞ (−1) n. ∑ 2 n2= n3n2++ ∞ cos(πnn) n o. ∑ n2= n! ∞ 1 p. ∑ 2 n2= nn(ln( )) Converge or diverge? Additional review problems. ∞ n3n22 +− a. ∑ 5 n2= 4n+ n− 1 ∞ n3n22 +− b. ∑ 6 n1= 4n+ n− 1 ∞ n5 c. ∑ n n1= 5 ∞ 1 d. ∑ . n=1 nn( +1) ∞ 1 e. ∑ 3 n=1 n ∞ n f. ∑ 3 n=1 nn+ 2 ∞ 2 g. ∑ n n=0 7 n ∞ −1 h. ( ) ∑ 2 n=1 n ∞ ⎛⎞11 i. ∑⎜⎟− n=1⎝⎠nn+1 ∞ 5 j. ∑ n=1 21n − ∞ 32n ∑ k. n=1 n! n ∞ ⎛⎞2n l. ∑⎜⎟ n=1⎝⎠51n − ∞ (1)− n−12n m. ∑ 3 n=1 31n + n ∞ ⎛⎞5 3 − n. ∑ ⎜⎟ n=0 ⎝⎠2 ∞ n ∑ o. n=1 n ∞ 1 p. ∑ −n n=11+ e ∞ 5n ∑ 3 q. n=1 n ∞ ∑cos(πn ) r. n=1 ∞ 1 s. ∑ 2 n2= nn( ln ) ∞ 3 ne− n ∑ t. n1= n ∞ ⎛⎞n u. ∑⎜⎟ n1= ⎝⎠n1+ ∞ 1 v. ∑ 3 n=1 n +1 ∞ n! w. ∑ n n=1 e n ∞ (−1n) ! x. ∑ n n2= n n ∞ (−1n) ! y. ∑ n2= 3n+ 2 n ∞ (−1n) ! z. ∑ n2= nn( + 1)! n ∞ (−−1n1) ( ) aa. ∑ 2 n2= 5n+ 2n− 1 ∞ cos(πn) bb. ∑ n2= n7+ n ∞ −12n cc. ( ) ∑ n n2= 21+ ∞ arctan(n) dd. ∑ 2 n2= 1n+ .