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Part of 2005 Grigore Moisil Urziceni

Problems(2)

lnln limit

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11/28/2019
Let be a sequence (an)n≥1 \left( a_n \right)_{n\ge 1} with a1>0 a_1>0 and satisfying the equality an=an+1−an+1+an, a_n=\sqrt{a_{n+1} -\sqrt{a_{n+1} +a_n}} , for all natural numbers n. n.
a) Find a recurrence relation between two consecutive elements of (an)n≥1. \left( a_n \right)_{n\ge 1} . b) Prove that lim⁡n→∞ln⁡ln⁡ann=ln⁡2. \lim_{n\to\infty } \frac{\ln\ln a_n}{n} =\ln 2.
Sequenceslimitsreal analysis
Rational product group under the operation (a,b)*(c,d) =(ac,ad+b)

Source:

11/28/2019
Define the operation (a,b)∘(c,d)=(ac,ad+b). (a,b)\circ (c,d) =(ac,ad+b).
a) Prove that (Q∖{0}×Q,∘) \left( \mathbb{Q}\setminus\{ 0\}\times\mathbb{Q} ,\circ \right) is a group. b) Let H H be an infinite subgroup of (Q∖{0}×Q,∘) \left( \mathbb{Q}\setminus\{ 0\}\times\mathbb{Q} ,\circ \right) that is cyclic and doesn't contain any element of the form (1,q), (1,q) , where q q is a nonzero rational. Show that there exist two rational numbers a,b a,b such that H={(an,b⋅1−an1−a)∣n∈Z} H=\left\{ \left.\left( a^n, b\cdot\frac{1-a^n}{1-a} \right)\right| n\in\mathbb{Z} \right\}
group theoryabstract algebra