By Andrews G E
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Extra info for Ramanujan Lost Notebook III. The Roger-Ramanujan Continued Fraction
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2. To this end, in order to prove that f( x) decreases, first note that the inequalities 2 x + 3 ≥ 0 and x 2 + 2 x − 6 ≥ 0 imply that x ≥ −(3/ 2) and x ≥ 7 − 1 or x ≤ − 7 − 1. 3 yields x ≥ 7 − 1. Therefore, the evaluation of ∆f ( x) can be carried out as follows ∆f ( x) = f ( x + ∆x) − f ( x) = 2( x + ∆x) + 3 − ( x + ∆x) + 3 − 2 x + 3 + x − 3 = 2( x + ∆x) + 3 − 2 x + 3 − ∆x = = ( 2( x + ∆x) + 3) 2 − ( 2 x + 3) 2 2( x + ∆x) + 3 + 2 x + 3 2 ∆x 2( x + ∆x) + 3 + 2 x + 3 − ∆x − ∆x 2 < ∆x − 1 2( 7 − 1) + 3 + 2( 7 − 1) + 3 1 = ∆x − 1 2( 7 − 1) + 3 1 1 < ∆x − 1 < ∆x − 1 < 0.
Once a switch from one level of concept awareness to another occurs, it turns out to be meaning generative. In such intellectual milieu, a teacher’s role becomes one of extreme complexity because of the need to survive ambiguity in a meaning-making environment that might take the power away from him or her (Crawford 1996). This suggests that the success of computational experiment approach is highly dependent on the competency of teachers. 2 serves as a single yet powerful illustration how mathematically competent teachers can turn this experiment into a meaning-making process for their students.
Through this variation, the numbers in column B would vary as well so that the arithmetic sequence with difference d ≥ 1 (and the first term equal to one) yields polygonal numbers of side d + 2 as its partial sums. 10 Computational Experiment as a Meaning-Making Process 17 Another basic principle that can be associated with the computational experiment approach is that human mental functioning (action) originates socially in the course of communication. Mediational means that human action usually employs are the products of social evolution and cultural development.