1.)

Determine whether the series is absolutely convergent, conditionally convergent or divergent. (-1)"(6n) n=1 V16n4 - 1 Part 1: Test _ lanl The series _ lanl converges Justification: O A. By the Limit Comparison Test, let ) bn = c where c = P = , and n = 1 n= 1 an lim bn O B. By the Divergence Test, lim |an| = 0 n-too Part 2: Conclusion The series _ an is conditionally cnovergent v Justification: O A. _ lan| converges, therefore _ an is absolutely convergent by definition. O B. _ an converges by the Alternating Series Test, because: i) {bn } is ultimately decreasing because the function f satifying f(n) = bn is decreasing on the interval ((6/32)^(1/4), infil ii) lim bn = 0 n-too O C. _ an diverges byy the Divergence Test, because lim an = n-tooDetermine whether the series is absolutely convergent, conditionally convergent or divergent. C(-1)-19n' + 3 n=1 3n3 + 15 Part 1: Test _ lanl The series _ lan converges v Justification: O A. By the Limit Comparison Test, let bn n=1 np where c = 1 p = 3/2 , and n=1 lan lim 1-too bn O B. By the Divergence Test, lim Jan) = n-too Part 2: Conclusion The series _ an is divergent Justification: O A. _ lan| converges, therefore _ an is absolutely convergent by definition. O B. _ an converges by the Alternating Series Test, because: i) {bn } is ultimately decreasing because the function f satifying f(n) = bn is decreasing on the interval ((13/3)^(1/3),infil ii) lim bn = 0 O C. _ an diverges byy the Divergence Test, because lim an =Determine whether the series is convergent or divergent. (\"may 2 The series '? v , Justication: (If more than one test is appropriate, pick the first applicable test in the list.) , cc) (' A. . . . . , . . J This is a Geometric Series of the form E our\" 1 where a : ,7' : , and its sum is 11:1 0 B. This is a Telescoping Series, lim 3,] : \"ADO O C. By the Divergence Test, n12:01G an = O D. By the Direct Comparison Test, an S I)\" with 2 bn : Z C(). C : and P : O E. By the Direct Comparison Test, an 2 b" where 26,. : Z c(#) where c : and p : O F. By the Limit Comparison Test. let Eb" : Z C() where c : ,p : . and O G. By the Alternating series test, i) {6,1} is ultimately decreasing because the function f satisfying flu) : b" is decreasing on the interval ii) lim bu : , \\ I . By the Integral Test, i) The function f satisfying n) = an is positive, continuous, and ultimately decreasing on the interval ii) [0 m) d1 : r. a 'J l' By the Ratio Test. lim \"+1 noo an O J. By the Root Test, Emmet {7 nl : (Enter \"DNE\" if divergent.) Determine whether the series is convergent or divergent. The series converges v. Justication: (If more than one test is appropriate, pick the first applicable test in the list.) . cc) (' A. . . . . , . . J This is a Geometric Series of the form E or\" 1 where a : ,7' : , and its sum is n:1 O B. This is a Telescoping Series, lim an = new 0 C. By the Divergence Test, lim 0.\" : noo O D. By the Direct Comparison Test, on g b\" with E b" : Z C(). c : and p : O E. By the Direct Comparison Test, an 2 bn where 21),. : Z c(%) where c : and p : O F. By the Limit Comparison Test. let Eb" : Z c( l where c : ,p : and L n}? O G. By the Alternating series test, i) {bu} is ultimately decreasing because the function f satisfying n) : bn is decreasing on the interval ii) 11m bn : TlDCI , l I . By the Integral Test, i) The function f satisfying u) : an is positive, continuous, and ultimately decreasing on the interval ii) IN f(a:) d1 : a " By the Ratio Test, 11131 "+1 nroo an : 4/125 O J. By the Root Test, Emum " lanl : (Enter \"DNE\" if divergent.) Determine whether the series is convergent or divergent. 6n ten n= 1 9n2 - 1 The series diverges Justification: (If more than one test is appropriate, pick the first applicable test in the list.) A. This is a Geometric Series of the form ) ar"- where a = , and its sum is (Enter "DNE" if divergent.) n=1 O B. This is a Telescoping Series, lim Sn = O C. By the Divergence Test, lim an = 0 O D. By the Direct Comparison Test, an c( ), c = and p = O E. By the Direct Comparison Test, an 2 bn where _ br = Ec( ) where c = and p = O F. By the Limit Comparison Test, let _ bn = Ec( m ) where c = , P = and lim an n too on O G. By the Alternating series test, i) {bn} is ultimately decreasing because the function f satisfying f(n) = bn is decreasing on the interval ii) lim by = O H. By the Integral Test, i) The function f satisfying f(n) = an is positive, continuous, and ultimately decreasing on the interval ") , f(z) da = O I. By the Ratio Test, lim Lan+1 n-700 an O J. By the Root Test, lim, too Vanl =Determine whether the series is convergent or divergent. i: 15tan'1(3n) 871 + 1 n:1 The series '? v , Justication: (If more than one test is appropriate, pick the first applicable test in the list.) 00 'J A' This is a Geometric Series of the form WWI where a : ,7" : , and its sum is 11:1 0 B. This is a Telescoping Series, lim 3,. : \"700 O C. By the Divergence Test, lim an : riPDO O D. By the Direct Comparison Test, on 31,1 with 2 bn : Z C(), c : and p : O E. By the Direct Comparison Test, an 2 by, where 211,, : Z c(#) where c : and p : O F. By the Limit Comparison Test. let Eb\" : Z C() where c : ,p : .and . an 11 _ : nrrgo bn O G. By the Alternating series test, i) {bu} is ultimately decreasing because the function f satisfying n) : b" is decreasing on the interval ii) hm b, = \"4'00 , \\ I . By the Integral Test, i) The function f satisfying u) = an is positive, continuous, and ultimately decreasing on the interval ii) [.0 x) d3: : r" a 'J I' By the Ratio Test, h'm "+1 naoo an Q J. By the Root Test. Emum " lanl : (Enter "DNE" it divergent.)