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 +...
Prove that for each natural number n 26 we have 2n 3 3 2" Use the above to prove that for each natural number n 2 6 we have (n +1)2 Hint: n24n +4-(n2 +2n +1) + (2n+3).] 2" Prove that for each natural number n 26 we have 2n 3 3 2" Use the above to prove that for each natural number n 2 6 we have (n +1)2 Hint: n24n +4-(n2 +2n +1) + (2n+3).] 2"
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
Question 16. A maximal plane graph is a plane graph G = (V, E) with n ≥ 3 vertices such that if we join any two non-adjacent vertices in G, we obtain a non-plane graph. (a) Draw a maximal plane graphs on six vertices. (b) Show that a maximal plane graph on n points has 3n − 6 edges and 2n − 4 faces. (c) A triangulation of an n-gon is a plane graph whose infinite face boundary is a...
Q9 (Approximation of π) (a) Show that 1/1 + t2 = 1 − t2 + t4 − ... + (−1)n−1 t 2n−2 + (−1)n t2n /1 + t2 for all t ∈ R and n ∈ N. (b) Integrate both side in (a), show that tan−1 (x) = x − x3/3 + x5 /5 − ... + (−1)n−1x 2n−1/ 2n − 1 + Z x 0 (−1)n t2n /1 + t2 dt. (c) Show that tan−1 (x) − ( x...
Q11 (Variant of Wallis product). For every integer n 2 0, we define Im r sin dx (a) Show that In+,-n+21n (b) Show that 0< 12n+2 S I2n+i < 12n- (c) Use (a) and (b), show that lim Pni1. (d) Repeatedly using (a), show that I2nl2 (e) Compute the limit lim (Historical remark: Q9 and Wallis Product was one of earliest approaches to 22n(n)! 24n(n!)4 (2n+1)!(2n)! 2-n(n!)2- n→oo v 2n+1(2n)!' 22n (n!)2 and then lim approximate π from rationals. This...
Does sigma (3n^2-n+1)/sqrt(n^7+2n^2+5) converge or diverge using limit comparison test.
A maximal plane graph is a plane graph G = (V, E) with n ≥ 3 vertices such that if we join any two non-adjacent vertices in G, we obtain a non-plane graph. A maximal plane graph is a plane graph G = (V, E) with n-3 vertices such that if we join any two non-adjacent vertices in G, we obtain a non-plane graph. (a) Draw a maximal plane graphs on six vertices b) Show that a maximal plane graph...
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
A maximal plane graph is a plane graph G = (V, E) with n ≥ 3 vertices such that if we join any two non-adjacent vertices in G, we obtain a non-plane graph. a) Draw a maximal plane graphs on six vertices. b) Show that a maximal plane graph on n points has 3n − 6 edges and 2n − 4 faces. c) A triangulation of an n-gon is a plane graph whose infinite face boundary is a convex n-gon...
Create a PDA that recognizes the language described. 1. {0n1m | n≠m} 2. {0n1m | m=2n} 3. {0^n1m | n≤m≤3n} 4. {w | w∈{0,1}∗,num of 0's in w=2(num of 1's in w)}