Some Applications of the Euler-Maclaurin Summation Formula
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1 International Mathematical Forum, Vol. 8, 203, no., 9-4 Some Applications of the Euler-Maclaurin Summation Formula Rafael Jakimczuk División Matemática, Universidad Nacional de Luján Buenos Aires, Argentina Abstract Let n be a fixed positive integer. In this article we obtain asymptotic formulae for N j n log j and N i= j (jn. For example if n = we obtain the asymptotic formulae, j log j = N 2 2 log N N N log N 2 log N C o(, i= and N j j e C i= where C is a constant. N N 2 2 N 2 2 e N2 4, Mathematics Subject Classification: A99 Keywords: Euler-Maclaurin summation formula, asymptotic formulae Introduction Bernoulli numbers are among the most distinguished and important numbers in all of mathematics. Indeed, they play a vital role in number theory. The Bernoulli numbers can be defined in the following way (see [], chapter V n B 0 = ( n B i =0 (n 2 i Using this relation we obtain the first Bernoulli numbers, namely B 0 =, B = /2, B 2 =/6, B 3 =0,B 4 = /30, B 5 =0.
2 0 R. Jakimczuk It is well known (see [], chapter V that if i 3 is odd then B i =0. The n-th Bernoulli polynomial is defined in the following way ( n B n (x = B j x n j. j j=0 The first few Bernoulli polynomials are B 0 (x =, B (x =x 2, B 2(x =x 2 x 6, B 3(x =x x2 2 x In the following theorem we establish the Euler-Maclaurin summation formula. Theorem. Let a<bbe integers and let m a positive integer. If f(x has m continuous derivatives on the interval [a, b], then b b m f(j = f(x dx ( i B ( i f (i (b f (i (a j=a a i= ( m b B m (x x f (m (x dx. ( m! a Proof. See [], chapter V. Let n be a fixed positive integer. formula P n (N = i= i n = N n n 2 N n n For example, we have P (N=2 N = N 2 We shall need the following well-known ( n B i N n i (N. i 2 N 2, P 2 (N = N 2 = N 3 3 N 2 2 N 6. P 3 (N = N 3 = N 4 4 N 3 2 N Preliminary Results Let us consider the function f(x =f (0 (x =x n log x, (2 where n is a positive integer. Let f (k (x be its k-th derivative. The following theorem holds.
3 Euler-Maclaurin summation formula Theorem 2. We have f (k (x =a k x n k log x b k x n k (k =,...,n, (3 where a k = n(n (n (k, (4 and k n(n (n (k b k =. (5 Besides f (n (x = n! x, (6 f (n2 (x = n! x. 2 (7 Proof. Clearly formulae (3, (4 and (5 are true if k =. This is the unique case if n =. Ifn 2, suppose that formulae (3, (4 and (5 are true for k such that k n. That is f (k (x = n(n (n (k x n k log x ( k n(n (n (k x n k. If we derive then we find that f (k (x = n(n (n (k (n kx n (k log x n(n (n (k x n (k ( k n(n (n (k (n k x n (k = n(n (n kx n (k log x ( k n(n (n k x n (k. That is, formulae (3, (4 and (5 are true for k. This proves formulae (3, (4 and (5. If k = n then equation (3 becomes (see (4 and (5 f (n (x =n! log x n n!. (8 Equations (6 and (7 are an immediate consequence of equation (8. theorem is proved. The We shall need the following integral x n log xdx= xn n log x (n 2 xn C. (9
4 2 R. Jakimczuk 3 Main Results Lemma 3. The integral is convergent. B n2 (x x dx (n (0 x2 Proof. Note that 0 x x <. On the other hand (see (, chapter V B n2 (0 = B n2 (. Consequently B n2 (x x is continuous and with period on the interval (,. Therefore there exist A>0 such that B n2 (x x A. Consequently we have B n2 (x x x 2 A x 2 (x. Now, the integral A x dx 2 is convergent. Therefore (comparison criterion the integral B n2 (x x dx (n x 2 is also convergent. Thus, the integral (0 converges absolutely and hence converges. The lemma is proved. The following theorem is our main theorem. Theorem 3.2 Let n be an arbitrary but fixed positive integer. The following asymptotic formula holds where j n log j = D n (N log N H n (NC n o( ( D n (N =P n (N( n B n (n, H n(n = N n (n 2 n B i b i N n (i. and C n is a constant depending of n. Note that D n (N and H n (N are polynomials in N of degree n.
5 Euler-Maclaurin summation formula 3 Proof. We have (see ( with m = n 2, (8, (6, (7, (9 and lemma 3. j n log j = N ( j n log j = x n log xdx B f (0 (N f (0 ( j=2 ( i B ( i f (i (N f (i ( ( n B n (n! (n! log N( n2 B n2 (n 2! ( n! N n! ( n (n (n 2 N B n2 (x x n n N N dx = log N x2 n (n 2 B f (0 (N ( i B i f (i (N( n B n (n log N C n o(, where C n is a constant. That is j n log j = N n n log N N n (n 2 B f (0 (N ( i B i f (i (N ( n B n (n log N C n o(. (2 Substituting (2 and (3 into (2 we obtain j n log j = N n n N log N n (n 2 2 N n log N ( i B ( i ai N n (i log N b i N n (i ( n B n (n log N C n o( (3 Since B = /2 (see the introduction. Equation (3 can be written in the form ( j n log j = A n (N( n B n log N H n (NC n o( (4 (n where A n (N = N n n 2 N n B i a i N n (i (5
6 4 R. Jakimczuk and H n (N = N n (n 2 n B i b i N n (i. (6 Since if i 2 then ( i B i = B i (see the introduction. Substituting (4 into (5 we find that (see the polynomial P n (N in the introduction A n (N = N n n 2 N n = N n n 2 N n n = N n n 2 N n n B i n (i (n(n (n (i 2N (n n (n (i 2 B i N n i ( n B i N n i = P n (N (7 i Equations (4, (5, (6 and (7 give equation (. The theorem is proved. Theorem 3.3 Let n be an arbitrary but fixed positive integer. The following asymptotic formula holds N j (jn Dn(N N Cn e, (8 ehn(n Proof. It is an immediate consequence of equation (. The theorem is proved. Remark 3.4 Note that equation (8 is a generalization of the Stirling s formula. Namely N j = N! e C N N/2 0, e N where in this case C 0 = log ( 2π. ACKNOWLEDGEMENTS. The author is very grateful to Universidad Nacional de Luján. References [] R. A. Mollin, Advanced Number Theory with Applications, Chapman and Hall/CRC, Taylor and Francis Group, Boca Raton, London, New York, 200. Received: August, 202
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