Question

Question 10 20 p1 Determine if the following series converges or diverges. If a test is used, please state the test determine and justify if any of the conditions for the test to be used are met, state any limits taken, and your final conclusion. If a series is used, state which one, and it is geometric or telescoping series and converges, find the sum

Answer 1

ni nan2+1 Solution! Let zan= & 2 1 - 1 1 2 4 n = ntn 21 I limit Comparison Test i Let Ean & Ebn be the Series of positive terms, an = non-rew finite Sach such that lim nto then both the series Ean and Eln converges or diverges together, P- Series Testi The series 2 1 converges it for and diverges of þé), Now let bn = - 3 an = 20- 1 x 3 = n(% - 1) bn nt_nati nu (1-2 +24) then = 2-in

lim an = lim 2-14 = Q = nonze40 and finite nee by limit Comparison test, both smies 1 Ean and Etu Ebu converges or converges or dives divereges together Beat Ebnt is convergent ty p-soei es test - with 10 = 30. San = 3 2n-1 is convergent. Sana na mht n = D E lend hits) Solution I let Ean= ln ( 2 ) Elo (n) - do(641) now, sequence of its partial aims is given by Si= as = log11) - log (2) 32 = a1 + a2 = (lon (1) - ln (2))+ ( ln (2) - ln (3) Sz= a + a2 + 9 = Eln(10-en (25) + (lon120-ln (31) + (ln(3) -enlus)

Sn = cyfa 2 tona an = (hes (1) - ln(2)) + (ln(2)- ln (3) toonit ... + ( ln (ngo - ln(n+1)) = lnce- ln (nt) how = -ln(n+1) as 3 an = ln(n Nel nal ) = lim Sn hooo = lim hao (- ln(n+1)) = -00 The as it series is not is the series convergent. of positive numbers, - The series is divergent © En Llorenja as a decreasing sequence of positive lan) is terms

Cauchy Condensation Test ! Sulopose ang is a decreasing tems. Then the series Ĕ an sequence of positive converges if and anly if the series § 2an converges. for the convergence Henue , by this test we check of Zanaan instead of Ean we have an=nleminge s ago to ago 32 - on poleg og - 29. 292. flog 292 => 39. Agne me larte ses now {2^ agn = Zn el los 22 let une leden 2 Von = 1

then lim un I'm - a nonzero nico An(21 g 2 flag 22 finite by limit comparison test Eun & Even both a Converges or diverges together. but Eln= { 1 2 is convergent by p-sener test. > {un= a ne niera converges → S (mm)2 Converges © Ean, let bu = 1 / 2 nas bn Ho nel then lim ana lim n = 1 = nonzero fmite. O - by limit comparison test both series Ean & Eba converges of diverges together. But Ebn= Et h is convergent by p-series test pz271 Ean= { 1 + is convergent.