Homework Answers
As per HomeworkLib policy I am liable to solve question 3. And to do that I used simple binomial theorem exapansion of x and y and finally put x=1 and y=-1 in that expasion to get the result.
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E3. Show that Σ(-1)k( ) = 0 for all positive integers n and k with 0-k-n...
Q3 (Prove that P∞ k=1 1/kr < ∞ if r > 1) . Let f : (0,∞) → R be a twice differentiable function with f ''(x) ≥ 0 for all x ∈ (0,∞). (a) Show that f '(k) ≤ f(k + 1) − f(k) ≤ f '(...
Q3 (Prove that P∞ k=1 1/kr < ∞ if r > 1) . Let
f : (0,∞) → R be a twice differentiable function with f ''(x) ≥ 0
for all x ∈ (0,∞).
(a) Show that f '(k) ≤ f(k + 1) − f(k) ≤ f '(k + 1) for all k ∈
N.
(b) Use (a), show that Xn−1 k=1 f '(k) ≤ f(n) − f(1) ≤ Xn k=2 f
'(k).
(c) Let r > 1. By finding...
(c) contrapositive positiv 2. (a) Prove that for all integers n and k where n >k>0,...
(c) contrapositive positiv 2. (a) Prove that for all integers n and k where n >k>0, (+1) = 0)+2). (b) Let k be a positive integer. Prove by induction on n that ¿ () = 1) for all integers n > k. 3. An urn contains five white balls numbered from 1 to 5. five red balls numbered from 1 to 5 and fiv
PLEASE SHOW ALL STEPS WITH EXPLAINATION Let m and n be positive integers and let k...
PLEASE SHOW ALL STEPS WITH EXPLAINATION Let m and n be positive integers and let k be the least common multiple of m and n. Show that mZ∩nZ=kZ.
Define σ as an element of S_z, where S_z is the collection of all permutations of all integers, by σ(k) = 3-k for all k element of all Integers. Find all the Orbits of σ.
Define σ as an element of S_z, where S_z is the collection of all permutations of all integers, by σ(k) = 3-k for all k element of all Integers. Find all the Orbits of σ.
Show that for all large positive integers n the sum 1/(n+1) + 1/(n+2) + 1/(n+3) +...
Show that for all large positive integers n the sum 1/(n+1) + 1/(n+2) + 1/(n+3) + ... + 1/(2n) is approximately equal to 0.693. I am trying to solve this problem by setting the sigma summation from k = n + k to 2n of 1/j to try to make a harmonic sum but is not working. I let j be n + k so it matches the harmonic sum definition of 1/k
DEFINITION: For a positive integer n, τ(n) is the number of positive divisors of n and...
DEFINITION: For a positive integer n, τ(n) is the number of
positive divisors of n and σ(n) is the sum of those divisors.
4. The goal of this problem is to prove the inequality in part (b), that o(1)+(2)+...+on) < nº for each positive integer n. The first part is a stepping-stone for that. (a) (10 points.) Fix positive integers n and k with 1 <ksn. (i) For which integers i with 1 <i<n is k a term in the...
Suppose that d = s and and positive integers m and n (a) Show that m/d...
Suppose that d = s and and positive integers m and n (a) Show that m/d and n/d are relatively prime ged(m, n) sm +tn for some integers (b) Show that if d = s'm + t'n for s', t' e Z, then s' = s kn/d for some k e Z.
3.) Let ak E R with ak > 0 for all k E N. Suppose Σ㎞iak...
3.) Let ak E R with ak > 0 for all k E N. Suppose Σ㎞iak converges. Show that Σί1bk (By definition, for a sequence (ck), we say liCkoo if, for all M ER with Hint: Show that there exists (Ni))ไ1 with N > Nj for all j E N, such that bk there exists a sequence (bk)k of real numbers such that lim converges = oo and M >0, there exists N E N such that ck > M...
2. Harmonic Mean Suppose all n data values in a data set are positive, with 0...
2. Harmonic Mean Suppose all n data values in a data set are positive, with 0 < y1S n < < Use the second derivative test from calculus to show that the value of the hypothesis h that minimizes the loss function L(h) = Σ (1-1)2 h i に1 is the harmonic mean of the data, defined as
Use the Binomial Theorem to show that Σ(-1): c(n, k)= 0 -0 Show transcribed image text
Use
the Binomial Theorem to show that
Σ(-1): c(n, k)= 0 -0
Q3 (Prove that P∞ k=1 1/kr < ∞ if r > 1) . Let
f : (0,∞) → R be a twice differentiable function with f ''(x) ≥ 0
for all x ∈ (0,∞).
(a) Show that f '(k) ≤ f(k + 1) − f(k) ≤ f '(k + 1) for all k ∈
N.
(b) Use (a), show that Xn−1 k=1 f '(k) ≤ f(n) − f(1) ≤ Xn k=2 f
'(k).
(c) Let r > 1. By finding...
(c) contrapositive positiv 2. (a) Prove that for all integers n and k where n >k>0, (+1) = 0)+2). (b) Let k be a positive integer. Prove by induction on n that ¿ () = 1) for all integers n > k. 3. An urn contains five white balls numbered from 1 to 5. five red balls numbered from 1 to 5 and fiv
DEFINITION: For a positive integer n, τ(n) is the number of
positive divisors of n and σ(n) is the sum of those divisors.
4. The goal of this problem is to prove the inequality in part (b), that o(1)+(2)+...+on) < nº for each positive integer n. The first part is a stepping-stone for that. (a) (10 points.) Fix positive integers n and k with 1 <ksn. (i) For which integers i with 1 <i<n is k a term in the...
Suppose that d = s and and positive integers m and n (a) Show that m/d and n/d are relatively prime ged(m, n) sm +tn for some integers (b) Show that if d = s'm + t'n for s', t' e Z, then s' = s kn/d for some k e Z.
3.) Let ak E R with ak > 0 for all k E N. Suppose Σ㎞iak converges. Show that Σί1bk (By definition, for a sequence (ck), we say liCkoo if, for all M ER with Hint: Show that there exists (Ni))ไ1 with N > Nj for all j E N, such that bk there exists a sequence (bk)k of real numbers such that lim converges = oo and M >0, there exists N E N such that ck > M...
2. Harmonic Mean Suppose all n data values in a data set are positive, with 0 < y1S n < < Use the second derivative test from calculus to show that the value of the hypothesis h that minimizes the loss function L(h) = Σ (1-1)2 h i に1 is the harmonic mean of the data, defined as
Use
the Binomial Theorem to show that
Σ(-1): c(n, k)= 0 -0
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