2. (12) Recall that Σί_1(21) andy n(n+1(2n+1). Com putey 2. (12) Recall that i- i' using...
Recall that, for all c, = n=0 cos(x) = § 4 (-1)",21 (2n)! and sin(x) = (-1)"..2n+1 (2n + 1)! N=0 n=0 If i is defined to have the property that i = -1, show that ei cos(2) + isin(x) for any real number r.
--Red In( - ) () -2 21 (+1) 2+1(n + m)} ml(m + 2n + 1) (1 - 1) de 22+1(n!) (2n +1) 8. By evaluating ac 2h +G ah where G(h) is the generating function for Legendre polynomials, show that 1 - 2 Σ (2n +1) Po (1 - 2ch + ha) Hence, or otherwise, prove that Pn(x) dx 2h 9. Given that {(2, 2) = ( 12h the hm-dh m>1. prove that 2am+ | 112,0)P.a)dt (m + n)...
Prove: without using l'hopital's rule. infinity 2n-1 ln(2) (2n-1) n infinity 2n-1 ln(2) (2n-1) n
3. (12 points) Consider the following sum: n Sn = {(i + 1)(i +2) i=0 (a) Use properties of summations to find a closed form expression for Sn. Simplify your answer into a polynomial with rational coefficients. Show your work, and clearly indicate your final answer. (b) Use weak induction to prove that your closed form works for every integer n > 0. Make sure you include all three parts, and label them appropriately!
Find the first four terms of the sequence given by the following. 21,=(-1)". 2n', n=1, 2, 3,... 0.000 OO Х 5
Help 2. = n(n+1)(2n+1) Cr=-=n(n+1) r(21-1) Given - and PT find in terms of e are integers (bn' + en + dn+e) Write your answer in the form a where a, b, possibly zero. Then b= d = and e = Part 2: 2
10) Use Theorem 4 to evaluate. n(n+1) 2 (=1 i = 10 pts 21-11 = n) 2,12 = n(n+1)(2n+1) E=1 43 = {*+1), 2 4 Theorem Iff is integrable on (a, b), then (x) dx = lim 8(xAx where and X; - a +i Ax (2x + 3 (2x + 5)dx
Some useful identities Using (2.3), we have n2n-1 n2n-1 + n(n 1)2"-2 non-1 + n(n-1)(n-2)2n-3 + 3n(n-1)2n-2 7n İfp-3 n- 22n(n1) if p 2 2"-3n2(n +3) if p3 Using (2.4) and (2.5) we have 0 ifpe(0, 1, ,n-1} Can you give combinatorial explanations for these identities?
Pt 1 pt 2 pt 3 pt 4 Please Answer every question and SHOW WORK! Determine whether the series n-1 Σ (2n)! 2". (2n! converge or diverge 1. both series converge 2. only series II converges 3. only series I converg es 4. both series diverge Determine whether the series 2! 1515.9 1-5.9-13 3! 4! 7m 1.5.9..(4n -3) is absolutely convergent, conditionally con- vergent, or divergent 1. conditionally convergent 2. absolutely convergent 3. divergent Determine which, if any, of the...
Help with any of these? Practice Problems 31 n* cos n 12 23. Σ㈠)"21/" 32n-1 n(ln n)3 n-2 24 + 5 -1 In n 25, Σ(-1)" 15 35 та і 36 (2n)" 16 26 17. Σ5n3nn Σ-π)" 7. k 27, 37 n-I E1 28. +1 10 k + 5 18 19 39 2.5.8(3n + 2)T ㄒㄧ- Σ(阪-1) 10 30 40 8m - 5 '고 (n + 1) (n-2) Practice Problems 31 n* cos n 12 23. Σ㈠)"21/" 32n-1 n(ln n)3...