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i=1 For en The purpose of this problem is to show that there exists a set...
10. Let dk -Ck*+1/2 exp(-k), where C is a strictly positive constant. At some point in the proof of Stirling's formula, we have that k! lim-= 1 and that (22n (n!)2 )2 (2nn+)2 (22n (dn)22 r π lim - Show that lim n→oo (dy.)"(2n+1) 2 10. Let dk -Ck*+1/2 exp(-k), where C is a strictly positive constant. At some point in the proof of Stirling's formula, we have that k! lim-= 1 and that (22n (n!)2 )2 (2nn+)2 (22n (dn)22...
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...
(2) Let r1 1 and -(-) 1 (n+1)2 = I+ur (a) Show that lim,, T, exists. (b) Prove that z, #t1 by induction and find the limit.
Part I: Show that (y − y ∗ 0 )(y − y ∗ 1 ). . .(y − y ∗ n ) = 5 n+1 2 n Tn+1(x), where x = y/5 Part II: It can be shown that there exists R > 0 such that |f (n) (y)| ≤ Rn for all y ∈ [−5, 5]. Assuming this, show that limn→∞ max{|f(y) − Pn(y)|, y ∈ [−5, 5]} = 0 Ij = COS Problem 1: Recall that the Chebyshev...
Problem 1: Recall that the Chebyshev nodes 20, 21, ...,.are determined on the interval (-1,1) as the zeros of Tn+1(x) cos((n + 1) arccos(x)) and are given by 2; +17 Tj = COS , j = 0,1,...n. n+1 2 Consider now interpolating the function f(x) = 1/(1 + x2) on the interval (-5,5). We have seen in lecture that if equispaced nodes are used, the error grows unbound- edly as more points are used. The purpose of this problem is...
Concept/Thinking (10 pts each) 1 1 1. Show lim no 1 1+ - + 2 + - Iv) Inn exists 3 n
please solve all and show steps thank you 1) Find the limit if it exists. x? - 7x+10 *+5 r + x - 30 a) lim tan x b) lim c) lim tan x In x x 0 2. Let f(x) = x° -18x' +4. a) Find the intervals on which f is increasing or decreasing, b) Find the local maximum and minimum values off. c) Find the intervals of concavity and the inflection points. d) Use the information from...
numerical methods 2+17), j = 0,1...... Problem 1: Recall that the Chebyshev nodes x0, 71,..., are determined on the interval (-1,1) as the zeros of Tn+1(x) = cos((n +1) arccos(x)) and are given by 2j +17 X; = cos in +12 Consider now interpolating the function f(x) = 1/(1+22) on the interval (-5,5). We have seen in lecture that if equispaced nodes are used, the error grows unbound- edly as more points are used. The purpose of this problem is...
class: numerical analysis I wish if it was written in block letter Sorry I can't read cursive = COS Problem 1: Recall that the Chebyshev nodes x4, x1,...,xy are determined on the interval (-1,1] as the zeros of Tn+1(x) = cos((n + 1) arccos(x)) and are given by 2j +10 Xj j = 0,1, ... 1 n+1 2 Consider now interpolating the function f(x) = 1/(1 + x2) on the interval (-5,5). We have seen in lecture that if equispaced...
1. This question is on probability a. Suppose that X is a normally distributed random variable, where X N (M, o). Show that E [cºX f (x)] = cºu+20oʻE [ f (x + 002)] where f is a suitable function and 0 € R is a scalar. Hint: Write X = 1 +o0; 0~ N (0,1) and calculate the resulting integral b. Consider the probability density function X>0 p(x) = { Az exp (-1.2-2) 10 x < 0 (>0) is...