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WebChallengeProblems_5E

WebChallengeProblems_5E - CHALLENGE PROBLEMS 1 Challenge...

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and therefore This gives an efficient way of computing the Bernoulli numbers and therefore the Bernoulli polynomials. (d) Show that and deduce that for . (e) Use parts (c) and (d) to calculate and . Then calculate the polynomials , , , , and . ; (f) Graph the Bernoulli polynomials for . What pattern do you notice in the graphs? (g) Use mathematical induction to prove that . (h) By putting in part (g), prove that (i) Use part (h) with and the formula for in part (a) to confirm the formula for the sum of the first cubes in Section 5.2. (j) Show that the formula in part (h) can be written symbolically as where the expression is to be expanded formally using the Binomial and each power is to be replaced by the Bernoulli number . (k) Use part (j) to find a formula for . |||| Chapter 6 1. A solid is generated by rotating about the -axis the region under the curve , where is a positive function and . The volume generated by the part of the curve from to is for all . Find the function .
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