LI YINGYING Theorem. Let Dn= k=1 k-log(n+1)(nEN)and y=limn-oo Dn be Euler constant. Then Tn: =?-Dn= 1(n+1)…(m+k) k t(1-t)…(k-1-t)dt(k>1) Furthermore 6(n+2)m(m+1)("m 之(+1)…(m+6)561(m+1)(m+7) For the numbers ak the referee provides the following table 1375 120a5=2046=504a7=168 He also points out that ak can be expressed in terms of Stirling numbers of the first kind s(k,j)as (-1)+y(k +. Our proof is completely an elementary calculation applying integral to estimate the sums. The method be ally applied to othe Acknowledgement The author is grateful to Professor Wang who enlightened her to write down this paper; also, she thanks the referee for valuable suggestions and for correcting printing mistakes refereNCeS 1. R. M. Young, Euler's Constant Math. Gazette 75 No 472(1991), 187-190 2. D W. De Temple, A quicker convergence to Euler's Constant, The Amer. Math. Monthly 100(1993),468-470 3. Rippon P L, Convergence with pictures, The Amer. Math. Monthly 93(1986), 476-478 4. D.W. De Temple and S.H. Wang, Half integer approximations for the partial sums of the harmonic series, J. Math lysis and Applic. 160(1991), 149-156 Department of Mathematics, Beijing Normal University, Beijing 100875, P.R. China ying-dd-Ii@263.net4 LI YINGYING Theorem. Let Dn = Pn k=1 1 k − log(n + 1) (n ∈ N) and γ = limn→∞ Dn be Euler constant. Then rn := γ − Dn = X∞ k=1 ak (n + 1)· · ·(n + k) , where a1 = 1 2 , ak = 1 k Z 1 0 t(1 − t)· · ·(k − 1 − t) dt (k > 1). Furthermore 1 6(n + 2)m(m + 1)¡m+n m ¢ < rn − Xm k=1 ak (n + 1)· · ·(n + k) < 1 6n(m + 1)¡m+n m ¢ . For the numbers ak the referee provides the following table a1 = 1 2 , a2 = 1 12 , a3 = 1 12 , a4 = 19 120 , a5 = 9 20 , a6 = 863 504 , a7 = 1375 168 , a8 = 33953 720 . He also points out that ak can be expressed in terms of Stirling numbers of the first kind s(k, j) as ak = (−1)k+1 k X k j=1 s(k, j) j + 1 . Our proof is completely an elementary calculation applying integral to estimate the sums. The method may be generally applied to other cases. Acknowledgement The author is grateful to Professor Wang who enlightened her to write down this paper ; also , she thanks the referee for valuable suggestions and for correcting printing mistakes. References 1. R.M.Young, Euler’s Constant, Math. Gazette 75 No.472 (1991), 187–190. 2. D.W. DeTemple, A quicker convergence to Euler’s Constant, The Amer. Math. Monthly 100 (1993), 468–470. 3. Rippon P L, Convergence with pictures, The Amer.Math.Monthly 93 (1986), 476–478. 4. D.W. DeTemple and S.H. Wang, Half integer approximations for the partial sums of the harmonic series, J. Math. Analysis and Applic. 160 (1991), 149–156. Department of Mathematics, Beijing Normal University, Beijing 100875, P.R. China ying-dd-li@263.net