Question

Find the convergence of the following series:

a.$\sum_{k=1}^{\infty }\frac{4k}{k^4+1}$ (Limit comparison test)

b. $\sum_{k=1}^{\infty }\frac{2k^2}{k^3+3k}$

c. $\sum_{k=1}^{\infty }\frac{k^2}{2k}$ (D'Alembert ratio test)

Answer 1

Using Limit Compersision test solution o Take the highest power ook in the numerator and the denominator - Ignoring any co-efficieus aand all other terms Ž AR B = 1 his This asenes resembles REIKAH ke K3 the convergent p-series, Now take the limit of the ratio of the Rth terms of two series 4K * Kto KAH ries lim K- O to is 4 and positive and since limit fo finite Convergent hence Convergent REI

6 Ø 2k² k=1 K3+3K Take the laso highest the denominator - agnon power of k in the namerator and any co-efficients and all other This series is they terms. s ak The K3+3K3 divergens hersmonis series Now take the limit of the ratio of the leth terms of two sen's 1 263 - 2162 lim K3+3K eim K) Kto K3+3K YK . line sa isi (14 314) 1_ 0 K3 = 2. since limit is finite and positive and our bench more series t is diversant hen § 212 es divergant divergent E, K3+3k is

Solution using D'Alembert ratio test ету, 6 ш чи then URH) = Et с 2 рет 2- +). - K42 л X 42 ики к24 +) е жt/ х 11 (ст. - К+ 2 к к urt (а - т HJ Ж -( 2 ) k- че К - 5 - Qiw K94 44 кою I lim kt@ E34 вс — 1+ + 3 + 73 Т+ +73 lt 2 + Yk2 - lim e->® 11 1+/к Долие 'я) схильехтело Qiu 1. o Цин | In uk - teen elm UK Кно fouls here.