欧拉常数γ及简单应用

Euler′s Constant γ and Its Simple Applications

调和级数∑∞n=11n是发散的,而极限limn→∞[∑nk=11k-lnn]却是收敛的,通常将极限值limn→∞[∑nk=11k-lnn]称为欧拉常数γ。欧拉常数γ存在性的证明有多种方法,例如,可利用函数不等式、几何直观(平面图形面积)、数项级数的收敛性、积分中值定理等方法。在微积分学中,欧拉常数γ有许多应用,如求某些数列的极限,某些收敛数项级数的和等。

The harmonics series ∑∞n=11n is divergent ,but the limit limn→∞[∑nk=11k-lnn]is convergent ,and the limit is called Euler's constant γ .There are many methods to prove the existence of the γ ,e.g.,using function inequality, direct observation of the geometry, the convergence of the number series, and theorem of mean . There are many applications of the Euler's constant in the calculus, e.g., how to get the limit of some number sequences and how to get the sum of some number series.