Let Σ an and Σ bn be series with positive terms such that lim(an/bn) = λ...

Let Σ a_n and Σ b_n be series with positive terms such that lim(a_n/b_n) = λ ∈ (0, ∞). Prove that the two series have the same behavior, that is they both converge or they both diverge to +∞.

Expert Solution

Colby Messinger answered 6 months ago

Let (xn) be a sequence with positive terms. (a) Prove the following: lim inf xn+1/ xn...

Let (xn) be a sequence with positive terms. (a) Prove the following: lim inf xn+1/ xn ≤ lim inf n√ xn ≤ lim sup n√xn ≤ lim sup xn+1/ xn . (b) Give example of (xn) where all above inequalities are strict. Hint; you may consider the following sequence xn = 2n if n even and xn = 1 if n odd.

(a) Let Λ = {λ ∈ R : 0 < λ < 1}. For each λ...

(a) Let Λ = {λ ∈ R : 0 < λ < 1}. For each λ ∈ Λ, let Aλ = {x ∈ R : −λ < x < 1/λ}. Find U λ∈Λ Aλ and ~U λ∈Λ Aλ respectively. (b) Let Λ = \ {λ ∈ R : λ > 1}. For each λ ∈ Λ, let Aλ = {x ∈ R : −λ < x < 1/λ}. Find U λ∈Λ Aλ and ~U λ∈Λ Aλ respectively.

Let {an} and {bn} be bounded sequences. Prove that limit superior {an+bn} ≦ limit superior {an}...

Let {an} and {bn} be bounded sequences. Prove that limit superior {an+bn} ≦ limit superior {an} + limit superior{bn}

Let (xn), (yn) be bounded sequences. a) Prove that lim inf xn + lim inf yn...

Let (xn), (yn) be bounded sequences. a) Prove that lim inf xn + lim inf yn ≤ lim inf(xn + yn) ≤ lim sup(xn + yn) ≤ lim sup xn + lim sup yn. Give example where all inequalities are strict. b)Let (zn) be the sequence defined recursively by z1 = z2 = 1, zn+2 = √ zn+1 + √ zn, n = 1, 2, . . . . Prove that (zn) is convergent and find its limit. Hint; argue...

5. Let n = 60, not a product of distinct prime numbers. Let Bn= the set...

5. Let n = 60, not a product of distinct prime numbers. Let Bn= the set of all positive divisors of n. Define addition and multiplication to be lcm and gcd as well. Now show that Bn cannot consist of a Boolean algebra under those two operators. Hint: Find the 0 and 1 elements first. Now find an element of Bn whose complement cannot be found to satisfy both equalities, no matter how we define the complement operator.

Let τ (n) denote the number of positive divisors of n and σ(n) denote the sum...

Let τ (n) denote the number of positive divisors of n and σ(n) denote the sum of the positive divisors of n (as in the notes). (a) Evaluate τ (1500) and σ(8!). (b) Verify that τ (n) = τ (n + 1) = τ (n + 2) = τ (n + 3) holds for n = 3655 and 4503. (c) When n = 14, n = 206 and n = 957, show that σ(n) = σ(n + 1).

2. Let X ~ Pois (λ) λ > 0 a. Show explicitly that this family is...

2. Let X ~ Pois (λ) λ > 0 a. Show explicitly that this family is “very regular,” that is, that R0,R1,R2,R3,R4 hold. R 0 - different parameter values have different functions. R1 - parameter space does not contain its own endpoints. R 2. - the set of points x where f (x, λ) is not zero and should not depend on λ . R 3. One derivative can be found with respect to λ. R 4. Two derivatives can...

lim[(tanx-1)/x] as x->0. lim[ (1+sin(4x))^cot(x)] as x -> 0 from the positive side. lim[ [(a^(1/x) +...

lim[(tanx-1)/x] as x->0. lim[ (1+sin(4x))^cot(x)] as x -> 0 from the positive side. lim[ [(a^(1/x) + b^(1/x))/2]^x] as x -> infinity

Test the series for convergence or divergence. ∞∑n=1(−1)nn4n Identify bn.

A force F = −F0 e ^−x/λ (where F0 and λ are positive constants) acts on...

A force F = −F0 e ^−x/λ (where F0 and λ are positive constants) acts on a particle of mass m that is initially at x = x0 and moving with velocity v0 (> 0). Show that the velocity of the particle is given by v(x)=(v0^2+(2F0λ /m)((e^-x/λ)-1))^1/2 where the upper (lower) sign corresponds to the motion in the positive (negative) x direction. Consider first the upper sign. For simplicity, define ve=(2F0 λ /m)^1/2 then show that the asymptotic velocity (limiting...

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