By Losang Thonden
The Library of Tibetan Works and data, based by means of His Holiness the Dalai Lama to maintain and foster the expansion of the Tibetan language and tradition, is proud to put up the current paintings, glossy Tibetan Language via Losang Thonden, Language learn student on the Library. A accomplished and updated textbook for instructing Tibetan throughout the medium of English, the fashionable Tibetan Language has been in response to the adventure won via the writer in his years of educating Tibetan to many Indian and international scholars.
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Extra info for Modern Tibetan Language, Volume I
Sample text
Then lim u(n) 0 n (24) In other words, the order of magnitude of odd-sided squares is less that the order of magnitude of rectangles at least one side of which is measured by an odd number. Proof. According to formulas (12) and (14) we have u(n) 2n(n 1)(2n 1) 2(2n 1) . 3n2 (n 1)(3n 1) 3n(3n 1) Therefore, 1 2 2(2n 1) 2 n 0 . lim u (n) lim lim n n 3n(3n 1) 3 n n(1 1 ) n This proves relation (24). The Multiplication Table from an Advanced Standpoint 33 In order to evaluate the ―smallness‖ of u(n), one can use a spreadsheet to investigate its behavior numerically prior to formal demonstration.
One can verify this fact by graphing the equation with the GC and finding the x-intercepts (of the resulting five vertical lines) through cursor pointing 3 Using the GC, one can check to see that the product of the (five real) roots of the equation x 5 10x 4 36x 3 56x 2 35x 6 0 is equal to -6. x n (1)n r . In the general case of the equation 46 Sergei Abramovich Remark 6. Relations (7) and (8), expressing the sum and the products of the roots of an equation through its coefficients, are known as Viète’s formulas for quadratic equation (6).
2n 1) 3(1 3 5 ... 2n 1) ... (2n 1)(1 3 5 ... 2n 1) (1 3 5 ... 2n 1)2 . 13 can be modified to display and add the highlighted numbers only, thereby providing a setting for experimental verification of formula (17). 13. Rectangles with odd dimensions on an odd size board. 6. COMPARING THE RATES OF GROWTH OF DIFFERENT FAMILIES OF GEOMETRIC FIGURES Formulas developed in this chapter for counting the number of geometric figures within the n n checkerboard and n n n Rubik’s cube demonstrated that such a number grows larger as n increases.