单调有界准则的推广与级数sum from n=1 to ∞ sin^m(an+b)/n~α的敛散性

A Generalization of the Monotone Bounded Principle and the Properties of Convergence and Divergence of Series sum from n=1 to ∞ sin^m(an+b)/n~α

利用致密性定理获得有界数列{y_n}收敛的一个充分条件:∨ε>0,■N∈Z+,使得当n>Z时,不等式yn-yn-1<ε恒成立。并发现任意项级数收敛的一个判定定理:如果级数sum from n=1 to ∞ a_n有界,且limn→∞a_n=0,则该级数收敛。由此获得:级数sum from n=1 to ∞ sin^(1+2s/t)=n/n~α收敛,其中s∈Z,t∈Z+,0<α≤1。并进行推广:如果s∈Z,t∈Z^+,0<α≤1,则级数sum from n=1 to ∞sin^1+2s/t)(an)/n~α收敛。再获得一个一般性结论:设有界函数f(n)满足0≤f(n)<M,且0<α≤1,则级数sum from n=1 to ∞sin(an)/n^af(n)收敛。同时利用确界定理得到:正项级数sum from n=1 to ∞sinn^(2s)n/n发散,其中s∈Z。并推广:正项级数sum from n=1 to ∞nsin^(2s)(an)/n发散,其中0<a≤π/2,s∈Z。利用数学归纳法获得:正项级数sum from n=1 to ∞ sin^(2s)(ann+b)/n发散,其中s∈Z,(a-2kπ)~2+(b-2lπ)~2>0,k,l∈Z。

By the compactness theorem, one sufficient condition for convergence of a bounded sequence {yn} is obtained that ∨ε〉0,■N∈Z＋,satisfies as n 〉N,{yn-yn-1｜〈εOne theorem for the convergence of any item series is found that if the series an is bounded,and limn→∞a_n=0,then this is convergent. Thus it is obtained that the series