Section 2

Convergence Exponent

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Problem 1.113

Show that

lim sup⁡m→∞log⁡mlog⁡rm=λ. \limsup _{m \rightarrow \infty } \frac{\log m}{\log r_m} = \lambda .
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Problem 1.114

Let x1,x2,x3,…,xm,…x_1, x_2, x_3, \ldots , x_m, \ldots be arbitrary real numbers, xm≠0x_m \neq 0. If there exists a positive distance δ\delta such that ∣xl−xk∣>δ\left|x_l-x_k\right|>\delta, l<kl<k, l,k=1,2,3,…l,k=1,2,3,\ldots, then the convergence exponent of ∣x1∣,∣x2∣,∣x3∣,…,∣xm∣,…\left|x_1\right|, \left|x_2\right|, \left|x_3\right|, \ldots , \left|x_m\right|, \ldots is at most 11.

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Problem 1.115

Let β\beta be larger than the convergence exponent of the sequence r1,r2,…r_1, r_2, \ldots. Then there exist infinitely many subscripts nn for which the n−1n-1 inequalities

r1rn<(1n)1β,r2rn<(2n)1β,…,rn−1rn<(n−1n)1β \frac{r_1}{r_n} < \left(\frac{1}{n}\right)^{\frac{1}{\beta }}, \quad \frac{r_2}{r_n} < \left(\frac{2}{n}\right)^{\frac{1}{\beta }}, \quad \ldots , \quad \frac{r_{n-1}}{r_n} < \left(\frac{n-1}{n}\right)^{\frac{1}{\beta }}

are satisfied.

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Remark.
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See problem 107.

Problem 1.116

Assume that the convergence exponent λ\lambda of the sequence r1,r2,r3,…,rm,…r_1, r_2, r_3, \ldots , r_m, \ldots is positive and 0<α<λ<β0<\alpha <\lambda <\beta. Then there are infinitely many subscripts such that the two types of inequalities

rμrn>(μn)1αfor μ=n−1,n−2,…,1, \frac{r_{\mu }}{r_n} > \left(\frac{\mu }{n}\right)^{\frac{1}{\alpha }} \quad \text{for } \mu =n-1, n-2, \ldots , 1, rνrn>(νn)1βfor ν=n+1,n+2,n+3,… \frac{r_{\nu }}{r_n} > \left(\frac{\nu }{n}\right)^{\frac{1}{\beta }} \quad \text{for } \nu =n+1, n+2, n+3, \ldots

hold simultaneously.

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Remark.
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See problem 109.