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Exercise 4.17. a) Find the least integer k such that 2n3 3n2 +3n 1 for all...
question 5 5. (a) Informally find a positive integer k for which the following is true:...
question 5
5. (a) Informally find a positive integer k for which the following is true: 3n + 1 < n2 for all integers n > k-4 (b) Use induction to prove that 3n +1 < n2 for all integers n 2 k. 6. Consider the following interval sets in R: B-4.7, E = (1,5), G = (5,9), M-[3,6]. (a) Find (E × B) U (M × G) and sketch this set in the-y plane. (b) Find (EUM) x (BUG)...
1 question) Arrange the following in the order of their growth rates, from least to greatest:...
1 question) Arrange the following in the order of their growth rates, from least to greatest: (5 pts) n3 n2 nn lg n n! n lg n 2n n 2 question)Show that 3n3 + n2 is big-Oh of n3. You can use either the definition of big-Oh (formal) or the limit approach. Show your work! (5 pts.) 3 question)Show that 6n2 + 20n is big-Oh of n3, but not big-Omega of n3. You can use either the definition of big-Omega...
2. Exercise 2. Consider the sequence (xn)n≥1 defined by xn = Xn k=1 cos(k) k +...
2.
Exercise 2. Consider the sequence (xn)n≥1 defined by xn = Xn k=1
cos(k) k + n2 = cos(1) 1 + n2 + cos(2) 2 + n2 + · · · + cos(n) n +
n2 . (a) Use the triangle inequality to prove that |xn| ≤ n 1 + n2
for all n ≥ 1. (b) Use (a) and the -definition of limit to show
that limn→∞ xn = 0.
Exercise 2. Consider the sequence (In)n> defined by cos(k)...
Problem 1 148pts] (1) I 10pts! Let P(n) be the statement that l + 2 +...
Problem 1 148pts] (1) I 10pts! Let P(n) be the statement that l + 2 + + n n(n + 1) / 2 , for every positive integer n. Answer the following (as part of a proof by (weak) mathematical induction): 1. [2pts] Define the statement P(1) 2. [2pts] Show that P(1 is True, completing the basis step. 3. [4pts] Show that if P(k) is True then P(k+1 is also True for k1, completing the induction step. [2pts] Explain why...
Consider the sequence defined as a[1] = 2; and a[k] = a[k-1]+2*k-1 for all positive integer...
Consider the sequence defined as a[1] = 2; and a[k] = a[k-1]+2*k-1 for all positive integer k >= 2; . Show that a[n] = 1+sum(2*i-1, i = 1 .. n); . Hint: Start with sum(2*i-1, i = 1 .. n);and use the recursive definition of the sequence.
Prove using mathematical induction that for every positive integer n, = 1/i(i+1) = n/n+1. 2) Suppose...
Prove using mathematical induction that for every positive integer n, = 1/i(i+1) = n/n+1. 2) Suppose r is a real number other than 1. Prove using mathematical induction that for every nonnegative integer n, = 1-r^n+1/1-r. 3) Prove using mathematical induction that for every nonnegative integer n, 1 + i+i! = (n+1)!. 4) Prove using mathematical induction that for every integer n>4, n!>2^n. 5) Prove using mathematical induction that for every positive integer n, 7 + 5 + 3 +.......
1. (Exercise 4.10, modified) Given a series Σ 1 ak with ak 0 for all k and lim Qk+1 k0oak we will prove that the series converges absolutely. (This is part of the ratio test sce the handout.) (a) Fix...
1. (Exercise 4.10, modified) Given a series Σ 1 ak with ak 0 for all k and lim Qk+1 k0oak we will prove that the series converges absolutely. (This is part of the ratio test sce the handout.) (a) Fix a valuc q with r <<1. Use the definition of r to prove that there exists a valuc N such that for any k 2 N. (b) Prove that Σο, laNIqk-1 converges, where N is the value from part (a)....
I need help on b-e. THANK YOU blem 3. Consider the following statement: 1 For all...
I need help on b-e. THANK YOU
blem 3. Consider the following statement: 1 For all n EN, 12 +22 +32 + ... +n? n(n+1)(2n +1) (a) Prove the statement () using mathematical induction. We use the term closed form expression to describe an algebraic expression that involves only a fixed amount of operations (i.e. that doesn't involving adding n terms). So for example, in the proposition above, the sum of n consecutive natural numbers (12 +22 + ... +...
I need help with number 3 on my number theory hw. Exercise 1. Figure out how...
I need help with number 3 on my number theory
hw.
Exercise 1. Figure out how many solutions x2 = x (mod n) has for n = 5,6,7, and then compute how many solutions there are modulo 210. Exercise 2. (a) Find all solutions to x2 +8 = 0 (mod 11). (b) Using your answer to part (a) and Hensel's Lemma, find all solutions to x2 +8 = 0 (mod 121). Exercise 3. Solve f(x) = x3 – x2 +...
show me all work for the problem i,ii,iii Exercise 1 (Sample size for estimating the mean)....
show me all work for the problem i,ii,iii
Exercise 1 (Sample size for estimating the mean). Let X1,...,x, be i.i.d. samples from some un- known distribution of mean u. Let X and S denote the sample mean and sample variance. Fix a E (0,1) and € >0. (i) Suppose the population distribution is N(uo?) for known op > 0. Recall that we have the following 100(1 - a)% confidence interval for : (1) Deduce that plue (x-Zalze in 2+ zarze...
question 5
5. (a) Informally find a positive integer k for which the following is true: 3n + 1 < n2 for all integers n > k-4 (b) Use induction to prove that 3n +1 < n2 for all integers n 2 k. 6. Consider the following interval sets in R: B-4.7, E = (1,5), G = (5,9), M-[3,6]. (a) Find (E × B) U (M × G) and sketch this set in the-y plane. (b) Find (EUM) x (BUG)...
2.
Exercise 2. Consider the sequence (xn)n≥1 defined by xn = Xn k=1
cos(k) k + n2 = cos(1) 1 + n2 + cos(2) 2 + n2 + · · · + cos(n) n +
n2 . (a) Use the triangle inequality to prove that |xn| ≤ n 1 + n2
for all n ≥ 1. (b) Use (a) and the -definition of limit to show
that limn→∞ xn = 0.
Exercise 2. Consider the sequence (In)n> defined by cos(k)...
Problem 1 148pts] (1) I 10pts! Let P(n) be the statement that l + 2 + + n n(n + 1) / 2 , for every positive integer n. Answer the following (as part of a proof by (weak) mathematical induction): 1. [2pts] Define the statement P(1) 2. [2pts] Show that P(1 is True, completing the basis step. 3. [4pts] Show that if P(k) is True then P(k+1 is also True for k1, completing the induction step. [2pts] Explain why...
Prove using mathematical induction that for every positive integer n, = 1/i(i+1) = n/n+1. 2) Suppose r is a real number other than 1. Prove using mathematical induction that for every nonnegative integer n, = 1-r^n+1/1-r. 3) Prove using mathematical induction that for every nonnegative integer n, 1 + i+i! = (n+1)!. 4) Prove using mathematical induction that for every integer n>4, n!>2^n. 5) Prove using mathematical induction that for every positive integer n, 7 + 5 + 3 +.......
1. (Exercise 4.10, modified) Given a series Σ 1 ak with ak 0 for all k and lim Qk+1 k0oak we will prove that the series converges absolutely. (This is part of the ratio test sce the handout.) (a) Fix a valuc q with r <<1. Use the definition of r to prove that there exists a valuc N such that for any k 2 N. (b) Prove that Σο, laNIqk-1 converges, where N is the value from part (a)....
I need help on b-e. THANK YOU
blem 3. Consider the following statement: 1 For all n EN, 12 +22 +32 + ... +n? n(n+1)(2n +1) (a) Prove the statement () using mathematical induction. We use the term closed form expression to describe an algebraic expression that involves only a fixed amount of operations (i.e. that doesn't involving adding n terms). So for example, in the proposition above, the sum of n consecutive natural numbers (12 +22 + ... +...
I need help with number 3 on my number theory
hw.
Exercise 1. Figure out how many solutions x2 = x (mod n) has for n = 5,6,7, and then compute how many solutions there are modulo 210. Exercise 2. (a) Find all solutions to x2 +8 = 0 (mod 11). (b) Using your answer to part (a) and Hensel's Lemma, find all solutions to x2 +8 = 0 (mod 121). Exercise 3. Solve f(x) = x3 – x2 +...
show me all work for the problem i,ii,iii
Exercise 1 (Sample size for estimating the mean). Let X1,...,x, be i.i.d. samples from some un- known distribution of mean u. Let X and S denote the sample mean and sample variance. Fix a E (0,1) and € >0. (i) Suppose the population distribution is N(uo?) for known op > 0. Recall that we have the following 100(1 - a)% confidence interval for : (1) Deduce that plue (x-Zalze in 2+ zarze...
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