Ray's New Primary Arithmetic for Young Learners by Joseph Ray

By Joseph Ray

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S ] Here are some series expansions that use Bernoulli numbers. Write procedures that evaluate these series (keeping terms through x15 ), using your symbolic system’s Bernoulli number function. Then compare the accuracy of your procedures with the results of direct function evaluations. (a) cot x = ∞ ∑ (−1)n 22n B2n 2n−1 x , (2n)! n=0 (b) tan x = ∞ ∑ (−1)n−1 22n (22n − 1)B2n 2n−1 x , (2n)! n=1 (c) csc x = ∞ ∑ (−1)n−1 2(22n−1 − 1)B2n 2n−1 x , (2n)! n=0 (d) ln cos z = ∞ ∑ (−1)n 22n−1 (22n − 1)B2n 2n x .

B) Using vectors, describe the locus of the points of the sphere. Solution: Letting r1 = (x1 , y1 , z1 ), (a) (x − x1 )2 + (y − y1 )2 + (z − z1 )2 = a2 (b) r = r1 + a, where a is a vector of magnitude a in an arbitrary direction. 7. Using formulas for vector components, show that the diagonals of a parallelogram bisect each other. Solution: Defining the parallelogram by vectors A, B, C, and D, defining vertices relative to a common origin, the condition that ABCD be a parallelogram is that D = A − B + C.

28 CHAPTER 2. INFINITE SERIES Solution: Write the binomial expansion (1 + x)−m/2 = = ∞ m m ∑ (− m 2 )(− 2 − 1) · · · (− 2 − n + 1) n x n! n=0 ∞ ∑ m(m + 2) · · · (m + 2n − 2) (−1)n xn n n! 2 n=0 ∞ ∑ (m + 2n − 2)!! (−1)n n = x . (m − 2)!! 2n n! 5. Write the function (1 − x)−n−1 as a series (keeping terms through x5 ). 3. 3 with results obtained from the series or Series command. In both languages it is necessary to convert the series into an ordinary expression (via convert or Normal). 6. 4. It suffices to check four values of n for each of three m values.

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