Question

1. [8 pts) Find the first four terms for each of the following sequences. the sequence converges or diverges and explain why. S(-1)"+12nº 5n²+6 Sn=1

Answer 1

Solution- 2 n © Piones Snt6 n21 - 2n 56²76 Now, the first four terms of this sequence are Us 281 ; Uz=zleme a 2x2 . 5(1+6 2 X 3 2x4 5(3)²+6 . 5(4) 76 86 - 43 Now, limuna lim 2nclinanse = lime nya. nyx Shto nyo wns where Un = 2 wn = 5 + 6 h But lim Una lim 2 zo O nya nyan and lim wn a lim (5+ M) = 5 tilim 14 2 2 =o) lim una lim 2 lim na him as an " 99 -SM+ เhe, note s 2n gee . Then Sequence { 24 and converges to Z ero. h is convergent

© S (-1) h*'2n 7 * nzi q 5ht6 Ini (htlan Let an= -(-ightlun, ny, I where 55r+6 2n una anal surto Now, the first four terms of this equation are 4 = 6 lt y = = 2 2 , x2 = (-192+1 K2=-42 = - 2 / 2 X3 =(-1) 3+1 432 43 = 7 / 8 s a tatluq 2-492 - 12/25 I we get the values of 41, 4₂, 43, ha from @7

Now we state Theorems 1: then every two theorems : If a sequence {uns converger tod, subsequence of {un} also converges to The orem 2: If the subsequences {uan} and {"anal} of a sequence un} converges to the same limit e then the sequence {uny is convergent and limunal. М-ы, але 2. = C) n = - Иә, . & Xzn-1 = (-320-17 Uanya Uanai Since, from @ we see that 1 lim una lim 2h - 20 nyx nyx 522+6 - Then, by the orem , lim Uanzo & lim Uan, 20 nya . lim na lim (-Uzn) - limuano nyx nyx nyot and lim dany a lim Uznal 20 mae nya nya

since, the subsequences {dan} & {danas} of a segnence 2003 converges to the same limit o. So, by the orem @ the sequence {en} is convergent and limano So the sequence {ihtlan 1. 15h1to and convergent to o 1 Сәптек-