Question

Calculus 2. Please use best handwriting as possible thank you!

5) Define Ža. converges to Se R in your own words (21 words or less): (4 pc 6) Given 21 19 17 15 43'85 127 169 write an expression for the term of the sequence and determine converg (61 7) Determine if the series Ž(21)" is convergent or divergent: ( 214 8) Evaluate lim 1+ 9) Determine if the series 2n is convergent or divergent: 2n24 + 3n-1

Answer 1

n=1 s] the series an converges to SEIR , it sequence of its partial sum {n} converges to S. 6) Here given sequence is, 45 Lei 21 So, 9,= 43 23-22 (1) 43 + 426b (1-1) 19 23-262) A2 = - 85 43 +42012-1 17 Az = 23-203) 127 43+ 42(3-1) ay 15 169 23- 2(4) 43+42(4-1) continuing in this way we can write, 23-2n an= 43+42(n-1) c6 so, aiuen sequence is fame {23.- is , 23-2m 43 +42(n-1) n=1

7) Here given series is (2ijn n! which 15 geometric series with first term a=21 and common ratio 8= 21 we Know geometric series aan con n=1 cenverges irf 101 21 and dirty diverges it? 10121 Hese 8 = 21 Hence nal 8 Here to find =) 101 = 1211 = 21 >| Geiven series 2 c20n diverges. ( 1+ )211 (i++) (i+b) lim hoo an we know, lim noo = 2in Hence, 21 11 lim nuo as here given series is, & 2n3. 2n24 n=1 +3n-1 So, Here en? ana 24 2n +3n-1 Now consider the series S nar n=1

The Here bna 21 we Know, P-series converges it no =1 P기 and diverges it psi So, swa converges as there P= 21 > 1 n21 How, lim nyoo 2n3 21 an bo lim naco 24 2n + 3n-1 n24 (2) lim nco 1 124 (2+ 3/23-24) (in3 n3.n21 n3+21 24 = n 11 2 =) limo no an bn lim n-co 2t 3n23-24 2 = - 2 + 3 2 2 an lim no 3.. - Yos a finite positive number limit compasision test, buth bn I, a finite M8 an So, by Nei cend 을 bn anverge ud diverge together. n=1 since, & Wish bn = converges, so n21 an 2n3 also converges. n=1 2n24+ 3n-1 n-1