对级数sum from n=1 to ∞(8nbn)的收敛性可用阿贝尔、犹利克雷判别法,而对其绝对收敛性却提文甚少;本文根据比较判别法直接研究级数sum from n=1 to ∞(a_nb_n)的绝对收敛性,并得出结果,用这结果判定了些级数的敛散性显得更加有效和方...对级数sum from n=1 to ∞(8nbn)的收敛性可用阿贝尔、犹利克雷判别法,而对其绝对收敛性却提文甚少;本文根据比较判别法直接研究级数sum from n=1 to ∞(a_nb_n)的绝对收敛性,并得出结果,用这结果判定了些级数的敛散性显得更加有效和方便。 一、定理及推论 1、定理:设sum from n=1 to ∞(a_n)是一无穷级数,{bn}是一序列。【i】若序列{bn}有畀且级数sum from n=1 to ∞(a_n)绝对收敛,则级数sum from n=1 to ∞(a_nb_n)绝对收敛;若序列{1/bn)有界且sum from n=1 to ∞|a_n|发散,则sum from n=1 to ∞n|a_nb_n|发散。 证明:假设sum from n=1 to ∞(a_n)绝对收敛且{b_n}有界,则存在正数M,使得|bn|【M,因此有|a_nb_n|≤M|a_n|因为sum from n=1 to ∞M|a_n|收敛,由比较判别法知sum from n=1 to ∞(a_nb_n)绝对收敛。 设sum from n=1 to ∞|a_n|发散且{1/b_n}有界,若sum from n=1 to ∞|a_nb_n|收敛,于(i)知sum from n=1 to ∞|a_nb_n|/|bn|=sum from n=1 to ∞|a_n|收敛,与条件矛盾,故级数sum from n=1 to ∞|a_nb_n|发散。展开更多
This note is devoted to the study of the absolute convergence of Bernstein polynomials. It is proved that for each x∈ , the sequence of the Bernstein polynomials of a function of bounded variation is absolutely su...This note is devoted to the study of the absolute convergence of Bernstein polynomials. It is proved that for each x∈ , the sequence of the Bernstein polynomials of a function of bounded variation is absolutely summable by |C,1| method. Moreover, the estimate of the remainders of the |C,1| sum of the sequence of the Bernstein polynomials is obtained.展开更多
文摘对级数sum from n=1 to ∞(8nbn)的收敛性可用阿贝尔、犹利克雷判别法,而对其绝对收敛性却提文甚少;本文根据比较判别法直接研究级数sum from n=1 to ∞(a_nb_n)的绝对收敛性,并得出结果,用这结果判定了些级数的敛散性显得更加有效和方便。 一、定理及推论 1、定理:设sum from n=1 to ∞(a_n)是一无穷级数,{bn}是一序列。【i】若序列{bn}有畀且级数sum from n=1 to ∞(a_n)绝对收敛,则级数sum from n=1 to ∞(a_nb_n)绝对收敛;若序列{1/bn)有界且sum from n=1 to ∞|a_n|发散,则sum from n=1 to ∞n|a_nb_n|发散。 证明:假设sum from n=1 to ∞(a_n)绝对收敛且{b_n}有界,则存在正数M,使得|bn|【M,因此有|a_nb_n|≤M|a_n|因为sum from n=1 to ∞M|a_n|收敛,由比较判别法知sum from n=1 to ∞(a_nb_n)绝对收敛。 设sum from n=1 to ∞|a_n|发散且{1/b_n}有界,若sum from n=1 to ∞|a_nb_n|收敛,于(i)知sum from n=1 to ∞|a_nb_n|/|bn|=sum from n=1 to ∞|a_n|收敛,与条件矛盾,故级数sum from n=1 to ∞|a_nb_n|发散。
文摘This note is devoted to the study of the absolute convergence of Bernstein polynomials. It is proved that for each x∈ , the sequence of the Bernstein polynomials of a function of bounded variation is absolutely summable by |C,1| method. Moreover, the estimate of the remainders of the |C,1| sum of the sequence of the Bernstein polynomials is obtained.
