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本文首先证明数列通项与前n项和的关系是一个充要条件,然后,应用它给出一类数列的一种初等求和方法。命题 S_n为数列{a_n}(n=1,2,3,…)的前n项和的充要条件为: 易知命题的必要性成立,现仅证充分性。证明由得∴命题的充分性成立。应用命题可给出下面一类数列的一种初等求和方法。 F(n)=sum from k=1 to n(1/k)f(k)r~(k-1) ①此处f(k)是含k的次数为m的任意多项式:
This article first proves that the relationship between a sequence general term and the first n terms is a necessary and sufficient condition. Then, it is used to give an elementary summation method for a sequence of numbers. The necessary and sufficient condition for the proposition that S_n is the first n terms of the series {a_n} (n=1, 2, 3,...) is: The necessity of the easy-to-know proposition is established and it is only sufficient for adequacy. The proof is established by the sufficiency of the proposition. Application propositions can give an elementary summation method for the following series of numbers. F(n)=sum from k=1 to n(1/k)f(k)r~(k-1) 1 where f(k) is any polynomial of degree m with k: