数学分析的对数p-级数与对数p-积分

Logarithmic p-serieses and logarithmic p-integrals in mathematical analysis

为了方便起见,将数项级数∑(lnn)^(k)/n^(p)称为对数p-级数,将反常积分∫_(c)^(+∞)(lnx)^(k)/x^(p) dx(c>1),∫_(a)^(b)[ln (x-a)]^(k)/(x-a)_(p)dx和∫_(a)^(b)[ln (b-x)]^(k)/(b-x)^(p)dx称为对数p-积分。对数p-级数与对数p-积分以不同的k和p的值出现在数学分析课程的各个知识环节中,收敛性随着具体参数k和p而不同,在数学分析和相关后续数学课程中有着很重要的地位。由于各类数学分析教材没有总结归纳出统一的判别其收敛性的公式化结论,导致对数p-级数与对数p-积分不能被学生熟练地掌握和使用。事实上,对数p-级数与对数p-积分的收敛性的判别完全可以公式化,根据k和p的值直接得出,这便弥补了数学分析课程在这些内容上的不足。

For convenience,the series ∑(lnn)k/npis called the logarithmic p-series,and the improper integrals ∫c+∞(lnx)k/xpdx(c>1),∫ab[ln (x-a)]k/(x-a)pdx and∫ab[ln (b-x)]k/(b-x)pdx are called the logarithmic pintegrals. They appear in various knowledge of mathematical analysis course with different values of k and p and different convergence,and play a very important role in mathematical analysis course and related subsequent mathematics courses. All kinds of mathematical analysis textbooks do not provide uniform formulaic conclusions about convergence of the logarithmic p-series and the logarithmic pintegrals,so that students cannot skillfully use them. In fact,their convergence can be formulated and can be gotten directly from the values of k and p. Therefore,these results complement the lack of such content in mathematical analysis courses.