Use the definition of 0 to show that 5n^5 +4n^4 + 3n^3 + 2n^2 + n 0(n^5).Use the definition of 0 to show that 2n^2 - n+ 3 0(n^2).Let f,g,h : N 1R*. Use the definition of big-Oh to prove that if/(n) 6 0(g{n)) and g(n) 0(h{n)) then/(n) 0(/i(n)). You should use different letters for the constants (i.e. don't use c to denote the constant for each big-Oh).
2) Prove that 1 + 3n < 4n for all n > 1. /5 Marks/
convergence of 2:58-(3n-1) 3.7-11.(4n-1) o0 2.5.8 (3n-1) (x 1) Find the radius of convergence of A. B. 4/3 C. 3/4 I),2 D. O E. 3/2 O F. 1/2 О н. 2/3 convergence of 2:58-(3n-1) 3.7-11.(4n-1) o0 2.5.8 (3n-1) (x 1) Find the radius of convergence of A. B. 4/3 C. 3/4 I),2 D. O E. 3/2 O F. 1/2 О н. 2/3
Determine whether the following series is absolutely convergent, conditionally convergent, or divergent. (–1)n-1((In n) 2n (3n+4)n • State the name of the correct test(s) that you used to reach the correct conclusion. • Show all work. • State your conclusion.
Please note n's are superscripted. (a) Use mathematical induction to prove that 2n+1 + 3n+1 ≤ 2 · 4n for all integers n ≥ 3. (b) Let f(n) = 2n+1 + 3n+1 and g(n) = 4n. Using the inequality from part (a) prove that f(n) = O(g(n)). You need to give a rigorous proof derived directly from the definition of O-notation, without using any theorems from class. (First, give a complete statement of the definition. Next, show how f(n) =...
3. Evaluate each of the following limits. 4n? - n +5 (a) an = (-1)","; (b) an= n+1 3n2+1 n n+1 (c) an= 5n (d) ann +1 n 3n (e) an=- () 4n = 5 - n+1 1.1" (g) an= (h) an= (-1)" 2 - 1 n
Order the following functions by asymptotic growth rate: 4n, 2^log(n), 4nlog(n)+2n, 2^10, 3n+100log(n), 2^n, n^2+10n, n^3, nlog(n) You should state the asymptotic growth rate for each function in terms of Big-Oh and also explicitly order those functions that have the same asymptotic growth rate among themselves.
4. Three forces of 2 N, 3 N and 4N act as shown below: Calculate the magnitude of the resultant and it's direction relative to the 2 N force. [6.24 N at 76.10°] 4N 60 3N 60° 2 N 1.5 m/s2 at 90° and b 2.6 m/s2 at 145° act at a [2.13 m/s2 at 0°] 5. Acceleration of a point. Draw a diagram and find a - b
12. Determine whether the following series converge or diverge. (a) (b) 2-nzn-1 4n n=0 n=1 4n (-1)n+1 loge n (c) (d) 7n + 1 n n=1 n=3 iM: M: Mį M8 sinn (e) ✓n n2 + 2 (f) n2 n=1 n=1 2n en (g) (h) Vn! n=1 n=1
PLEASE SHOW WORK Question 3 Use mathematical induction to prove 3+7+11+ ... +(4n – 1) = n (2n + 1). • Show P1 is true. • Assume Pk is true. • Show Pk+1 is true.