Section 5

Sequences of Complex Numbers, Continued: Transformation of Sequences

[7/100%]
Show
LaTeX
Problem 3.44

Prove the easier part of the above mentioned proposition: If the conditions (1) and (2) are satisfied the array preserves convergence. [Problems 66 and 80 of Part I.]

?
Problem 3.45

What conditions must the series u0+u1+u2+⋯+un+⋯u_{0} + u_{1} + u_{2} + \cdots + u_{n} + \cdots satisfy in order that its Cauchy product [Problem 34 of Part I, Problem 23 of Part II, Part VIII Chapter 1 5] with any convergent series v0+v1+v2+⋯+vn+⋯v_{0} + v_{1} + v_{2} + \cdots + v_{n} + \cdots results in a convergent series

u0v0+(u0v1+u1v0)+(u0v2+u1v1+u2v0)+⋯+(u0vn+u1vn−1+⋯+un−1v1+unv0)+⋯? u_{0}v_{0} + (u_{0}v_{1}+u_{1}v_{0}) + (u_{0}v_{2}+u_{1}v_{1}+u_{2}v_{0}) + \cdots + (u_{0}v_{n}+u_{1}v_{n-1}+\cdots +u_{n-1}v_{1}+u_{n}v_{0}) + \cdots ?
?
Problem 3.46

What condition must the series u1+u2+⋯+un+⋯u_{1}+u_{2}+\cdots +u_{n}+\cdots satisfy in order that its Dirichlet product [Part VIII, Chapter 1, 5] with any convergent series v1+v2+⋯+vn+⋯v_{1}+v_{2}+\cdots +v_{n}+\cdots results in a convergent series

u1v1+(u1v2+u2v1)+(u1v3+u3v1)+⋯+∑l∣nulvnl+⋯? u_{1}v_{1} + (u_{1}v_{2}+u_{2}v_{1}) + (u_{1}v_{3}+u_{3}v_{1}) + \cdots + \sum _{l\mid n} u_{l}v_{\frac{n}{l}} + \cdots ?
?
Problem 3.47

The sequence of factors γ0,γ1,γ2,…,γn,…\gamma_{0}, \gamma_{1}, \gamma_{2}, \ldots , \gamma_{n}, \ldots turns any convergent series a0+a1+a2+⋯+an+⋯a_{0}+a_{1}+a_{2}+\cdots +a_{n}+\cdots into a convergent series

γ0a0+γ1a1+γ2a2+⋯+γnan+⋯ \gamma _{0}a_{0} + \gamma _{1}a_{1} + \gamma _{2}a_{2} + \cdots + \gamma _{n}a_{n} + \cdots

if and only if the series

∣γ0−γ1∣+∣γ1−γ2∣+⋯+∣γn−γn+1∣+⋯ \left|\gamma _{0}-\gamma _{1}\right| + \left|\gamma _{1}-\gamma _{2}\right| + \cdots + \left|\gamma _{n}-\gamma _{n+1}\right| + \cdots

converges.

?
Problem 3.48

[Cf.G. Polya, Problem: Arch. Math. Phys. Ser. 3, Vol. 24, p.282 (1916). Solved by S. Sidon: Arch. Math. Phys. Ser. 3, Vol. 26, p.68 (1917).] Set

cu0=z0,u0+u1+⋯+un−1+cun=zn,n=1,2,3,…, cu_{0} = z_{0}, \quad u_{0}+u_{1}+\cdots +u_{n-1}+cu_{n} = z_{n}, \quad n = 1, 2, 3, \ldots , u0+u1+⋯+un−1+un=wn,n=0,1,2,…, u_{0}+u_{1}+\cdots +u_{n-1}+u_{n} = w_{n}, \quad n = 0, 1, 2, \ldots , ∑n=0∞unζn=U(ζ),∑n=0∞znζn=Z(ζ),∑n=0∞wnζn=W(ζ). \sum _{n=0}^{\infty } u_{n}\zeta ^{n} = U(\zeta ), \quad \sum _{n=0}^{\infty } z_{n}\zeta ^{n} = Z(\zeta ), \quad \sum _{n=0}^{\infty } w_{n}\zeta ^{n} = W(\zeta ).

The existence of lim⁡n→∞(u1+u2+⋯+un−1+cun)=α\lim_{n\to \infty }(u_{1}+u_{2}+\cdots +u_{n-1}+cu_{n}) = \alpha implies the existence of lim⁡n→∞(u1+u2+⋯+un−1+un)=α\lim_{n\to \infty }(u_{1}+u_{2}+\cdots +u_{n-1}+u_{n}) = \alpha in two cases only: if c=0c = 0 or if Re(c)>12\mathrm{Re}\left(c\right) > \frac{1}{2}, but not if Re(c)≤12\mathrm{Re}\left(c\right) \leq \frac{1}{2}, c≠0c \neq 0.

?
Problem 3.49

Let u0,u1,u2,…,un,…u_{0}, u_{1}, u_{2}, \ldots , u_{n}, \ldots be arbitrary complex numbers. For what values of cc does the existence of

lim⁡n→∞(un+cu0+u1+⋯+unn+1) \lim _{n\to \infty }\left(u_{n} + c\frac{u_{0}+u_{1}+\cdots +u_{n}}{n+1}\right)

imply the existence of lim⁡n→∞un\lim_{n\to \infty } u_{n}?

?
Problem 3.50

If the Dirichlet series

a11−s+a22−s+a33−s+⋯+ann−s+⋯ a_{1}1^{-s} + a_{2}2^{-s} + a_{3}3^{-s} + \cdots + a_{n}n^{-s} + \cdots

converges for s=σ+iτs = \sigma + i\tau, σ,τ\sigma , \tau real, σ>0\sigma > 0, then

lim⁡t→1−0(1−t)σ(a1t+a2t2+a3t3+⋯+antn+⋯ )=0.[Problem 92 of Part I.] \lim _{t\to 1-0} (1-t)^{\sigma }(a_{1}t + a_{2}t^{2} + a_{3}t^{3} + \cdots + a_{n}t^{n} + \cdots ) = 0. \quad [\text{Problem 92 of Part I.}]
?