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We study the growth at the golden rotation number ω = ( 5 − 1 ) / 2 of the function sequence P n ( ω ) = ∏ r = 1 n | 2 sin ⁡ π r ω | . This sequence has been variously studied elsewhere as a skew product of sines, Birkhoff sum, q-Pochhammer symbol (on the unit circle), and restricted Euler function. In particular we study the Fibonacci decimation of the sequence P n , namely the sub-sequence Q n = | ∏ r = 1 F n 2 sin ⁡ π r ω | for Fibonacci numbers F n , and prove that this renormalisation subsequence converges to a constant. From this we show rigorously that the growth of P n ( ω ) is bounded by power laws. This provides the theoretical basis to explain recent experimental results reported by Knill and Tangerman (2011) [10] . [ABSTRACT FROM AUTHOR]

Abstract: Analyzing several classical tests for convergence/divergence of number series, we relax the monotonicity assumption for the sequence of terms of the series. We verify the sharpness of the obtained results on corresponding classes of sequences and functions. [ABSTRACT FROM AUTHOR]