广义积分与无穷级数收敛定理

A Theorem for the Convergence of Generalized Integral and Infinite Series

指出若ｆ（ｘ）＝Σ∞ｎ＝０ａｎｘｎ，且收敛半径不小于１，则广义积分∫１０ｆ（ｘ）（１－ｘ）ｐｄｘ收敛的充要条件是无穷级数Σｎｐ－１ａｎ１ｎｎ收敛．

It is proved in the presented work that for the function f(x)=Σ∞ n=0a nx n with convergent radius not less than 1,the condition for the convergence of the genealized integral ∫ 1 0f(x)(1-x) pd x is that the infinite series Σ n p-1 a n ln n converges.