Limit of a SeriesDate: 03/02/2003 at 13:39:58 From: Hilaf Hasson Subject: A Limit Problem What is the limit of the following series: a(n) = ((1^n) + (2^n) + ... + (1998^n))^(1/n) Thank you for your time. Date: 04/01/2003 at 23:05:51 From: Doctor Nitrogen Subject: Re: A Limit Problem Hi, Hilaf: For your series: a(n) = ((1^n)+(2^n)+...+(1998^n))^(1/n) you can first compute the value of (A) (1^n)+(2^n)+...+(1998^n) by using something called Bernouilli Polynomials B_n(x). That's to be read 'B subscript n of x', or B(x) but with the 'B' subscripted with n. Then you will be able to compute a(n) = ((1^n)+(2^n)+...+(1998^n))^(1/n) for any positive integer value of n. Let kCi denote the combinatorial symbol. Bernouilli polynomials are found from SUM(i = 0 to k)kCi(B_k-i times x^i). The first four Bernouilli polynomials are: 1. B_0(x)= 1, 2. B_1(x)= x - 1/2, 3. B_2(x) = x^2 - x + 1/6, 4. B_3(x) = x^3 - (3/2)x^2 + x/2, etc. The coefficients appearing in the polynomials are called Bernouilli numbers and they are given by computing SUM(i = 0 to k = k-1)kCi(B_i) = 0. There is a formula you can use to compute (A): SUM(x = 1 to k)x^n = [B_n+1(k + 1) - B_n+1(0)]/n+1. using this formula, (A) is found: (B) (1^n)+(2^n)+...+(1998^n) = [B_n+1(1999) - B_n+1(0)]/n+1, where B_n+1(1999) and B_n+1(0) are the (n+1)-st Bernouilli polynomial with arguments x = 1998 + 1 = 1999 and x = 0, respectively. You can use (B) to calculate (A), then take the n-th root (1/n) for any positive integer value of n. If you are interested in further research into Bernouilli polynomials and Bernouilli numbers, there is an extraordinary paper on this topic: "A New Approach to Bernouilli Polynomials," by D.H. Lehmer, American Mathematical Monthly, vol. 95, #10, December, 1988, p. 905. You can also find information on Bernouilli polynomials/numbers in: "Handbook of Mathematical Functions," Abramowitz, Stegun, editors. I hope this helped answer the questions you had concerning your mathematics problem. - Doctor Nitrogen, The Math Forum http://mathforum.org/dr.math/ |
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