数学与应用数学外文中英文翻译.doc

本科生毕业设计（论文） 外文翻译 A Discussion on a Limit Theorem and Its Application Abstract: This paper proposes that a limit theoremcan help to solve a specific limit problemof sum formula and that some limit of product formula can also be solved by exploiting the feature of logarithm function. Keywords: limit theorem; sumformula; product formula Incalculus,we will usually solve a specific limit problem of sum formula But this sum formula can’t sum directly, and it can’t change into some kinds of function’s integral sum. So it is hard to work out its limit , for solve this problem. This paper’s proposes is that a limit theorem can help to solve this limit problem of sum formula and that some limit of product formula can also be solved by logarithm function. Theorem1 Let (a) f be differentiable at x=0 and f (0) =0,(b) g be integrable for x∈[a, b]. We have Proof 　By the (a), for every thereis a 0 such that implies . Then by the (b), there exists a real number M0 such that | g(x)| ≤M for x∈[a,b] and there is a 0 such that‖T‖implies Let ,so when‖T‖δ, we get and therefore We note the preceding argument was based on the assumption that f (0) =0. For the case that f (0) ≠0. We can show that for f (0) 0 and Let f (x) =x then theorem 1 has become This is definition of definite integral , and by logarithm function we get Corollary2 If f be differentiable at x=0 and f (0) =1 and g be integrable for x into [a,b] then we have In practical is usually divide [0,1] into n parts, and choose (k=1,2, …, n). Corollary3　Let f be differentiable at x=0 and g be integrable for x into [0,1] , then we have If f (0) =0, we have If f(0) =1, we have Proof 　By that theorem1 and logarithm function, we get Example1　Evaluate each of the following: Solution　(a) Rewrite the sum in the equivalent form So that by theorem1, （b）Rewrite the sum in the equivalent form So that by theorem1, So that by theorem1, （d）Let f(x) =sinax and g(x) =x. Then So that by theorem 1, So that by theorem 1, Example2