Question

3) Do the following series diverge, absolutely converge, or conditionally converge? Write which test you use for each part.

(9 points each) (Hint: use L'Hopital's Rule for part b)

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Answer 1

K=2 Ratco test :- ( 1) k (akt1 -akya det an akt k K-1 kte-2 okti 1) (K- L = antl lum lem Kyo Kdo ( 1) CU len KY We 1 - HK K 700 1 E lem 1-2). 1 - 1 decreasehg so, Ratco test breets as L=1 Now, Alternating se 1 seroes test! Here, 1, is decreasing as, for, k-2, 2 *3. Lo vence et es decreasing for k=2 1 Kol Ke 3 R-1 and om 1 - son 1 KO (K-1) ky 1- convergent. so, et as cor, o us comme & 2 art z (2) is absolutely Now, un convergent K=2 2 os absolutely convergent et es convergent det's Cheen it by integral test.

set f (x) - 2 - 1 12-11 then lem E 1 t 7D 2-1 = won log 16-11 - log (1) ( log(1) = 0. 2 um It-1) tuo Hence, The series M8 (-2) only conditconally convergent . K-2 K-2 1332 Roto test Aterpretenee = (-1) ktk here, al= (-1) k 2K , aut Kl. k Now a kth lim | (-1)kte I kth (1)k ak & com | (-1). 2k lim (+1). 2 Kykte 1 k.70 14 12700 ktz es 1-21- 2 1 so, This serves as dororgent. Note: test it L<2 then only series for ratco converges