2. Prove that lim (-1)"+1 0. 72-00 n 2n 3. Prove that lim noon + 1...
3. Find: 7T (1) lim n sin (3) lim arcsin G n-00 n 100 COS- n n00 n 1 (4) lim (1+ (6) lim (n+1 n) n- 3n n-00 1/n (2) lim arctann Vn2 - 1 (5) lim 2n In (1+) (8) lim n (11) lim n+ n 2+1 (14) lim n- 75n+2 (9) lim (nt n-00 700 n->00 n (7) lim V (Vn+1- Vn) (-1)"n (10) lim n+ on+1 (13) lim (3" +5")1/n sinn (12) lim arctan 2n 2n...
m2 2. Prove that lim -+0n3 + 1 -=0. 3 5 100 3n2 + 2n - 1 3. Prove that lim = 5n2 +8 cos(n) 4. Prove that lim = 0. n-700 m2 + 17 5. Prove that lim (Vn+1 - Vn) = 0 Hint: Multiply Vn+1-vñ by 1 in a useful way. In particular, multiply Vn+1-17 by Vn+1+vn
2n 3. Prove that lim n+on+ 1 2.
9 – in 5. Prove that lim n+ 8 + 13n -7 13
2. Prove that lim (-1)"+1 = 0. n-00 n
1. What does it mean for a sequence {a} to converge to a € R? State the definition. (-1)n+1 2. Prove that lim = 0 n 2n 3. Prove that lim +0n + 1 = 2 80 4. Prove that lim +-+V5n 9 - 7 5. Prove that lim 108 + 137 13
Find the interval of convergence of the power series: 5) 00 2n -(4x – 8)" n n=1 E (n + 1)(x - 2)" (2n + 1)! n=0 7) 00 w n(x + 10)" (2n)! n=0
Prove that P2n(0)= (-1)n ((2n-1)!!/(2n)!!) using the generation function and a binomial expansion. Show that (sqrt(pi)(4n-1)/(2gamma(n+1)gamma(3/2-n))=(-1)n-1((2n-3)!!/(2n-2)!!)(4n-1)/2n
Please and thank you! purpose of this project is to develop Wallis's formula. Forn 0.1,2,.., define The 6. Prove that 2e12 12 3-3-5-5-7.7-(2n (2n (2n 1)m 2-2.4.4-6-6(2n)(2n) 2 Parts 5 and 6 yield Wallis's formula: 2-2.4-4-6-6(2n)(2n) niin 1-3-3-5-5-7-7 (2n-I)(2n-1)(2n + 1) = 2. lim Wallis's formula gives as an infinite product, defined as the limit of partial products, in much the same way we defined the infinite sum as the limit of partial sums. If you continue your study of...
ad cal 2 . Problem 8, (10 pts.) Prove that on the interval [0,0.8) -2n lim dx Problem 9, (10 pts.) na(1-z)". Let fn (z) Prove that . Problem 10, (10 pts.) Using Method of mathematical induction prove that: If function u(x) is such that a,--u then a ,u u, 2n1 . Problem 8, (10 pts.) Prove that on the interval [0,0.8) -2n lim dx Problem 9, (10 pts.) na(1-z)". Let fn (z) Prove that . Problem 10, (10 pts.)...