由买买提看人间百态

boards

本页内容为未名空间相应帖子的节选和存档,一周内的贴子最多显示50字,超过一周显示500字 访问原贴
Science版 - 一个极限问题
相关主题
Re: 数列问题Re: Summation of Sequence
淋雨的概率problem: travelling wave in a ring network
Re: [转载] 高手们帮个忙Re: A set problem
Re: what is 经典力学中的Hamilton方程?help with numerical integral transformation
Re: 谁给俺讲讲Gauss-Chebyshev quadrature formulaRe: 求教相速度和群速度
Re: 请教积分,急!Re: 帮人问个数学问题
Re: how to compute the expected value in Mathematica?我的答案Re: 淋雨的概率
帮我想想这个数学展开吧。Re: is it a joke or a surprise? MgB_2 is a high temperature supercondu
相关话题的讨论汇总
话题: lim话题: infty话题: sum话题: 无穷小
进入Science版参与讨论
1 (共1页)
f**n
发帖数: 401
1
Define A_n=\sum_{i=1}^n a_i, B_n=\sum_{i=1}^n b_i in which a_i and b_i are
both 无穷小量 or equivalently:
\lim_{n->\infty}a_i=\lim_{n->\infty}b_i=0
Moreover for the same i, a_i and b_i are 等价无穷小 or equivalently:
\lim_{n->\infty}a_i/b_i=1, 1<=i<=n
Now we want to prove:
\lim_{n->\infty}A_n=\lim_{n->\infty}B_n if either of the limits exists.
The above equation seems intuitively true...to me. But
it seems that the proof requires the exchange of \lim and \sum.
I am not sure the requirements for this exch
I***e
发帖数: 1136
2

I don't think the conclusion is right. Take a1=1 and ai=0 for i>1. Take bi=0
for all i.
THen An = 1 and Bn=0 for all n.
How can lim An = lim Bn ?
I guess you meant An= sum_{i=n}^infty ai. If so, you have to also have some
extra conditions like ai, bi are both positive sequences. Then you can use the
discrete version of BCT ( Bounded convergence theorem )
Hope this helps.
icare

【在 f**n 的大作中提到】
: Define A_n=\sum_{i=1}^n a_i, B_n=\sum_{i=1}^n b_i in which a_i and b_i are
: both 无穷小量 or equivalently:
: \lim_{n->\infty}a_i=\lim_{n->\infty}b_i=0
: Moreover for the same i, a_i and b_i are 等价无穷小 or equivalently:
: \lim_{n->\infty}a_i/b_i=1, 1<=i<=n
: Now we want to prove:
: \lim_{n->\infty}A_n=\lim_{n->\infty}B_n if either of the limits exists.
: The above equation seems intuitively true...to me. But
: it seems that the proof requires the exchange of \lim and \sum.
: I am not sure the requirements for this exch

f**n
发帖数: 401
3
Yes. Sorry about that.
Let me try to make it clearer:
Actually I am trying to emulate an integral by a summation:
You can imagine there is a interval(x axis) and I divide the interval into
n smaller intervals. After this decomposition, we index all intervals, let's
say, interval 1,2,...,n. For interval i, a_i and b_i are defined.
So if n->\infty, both series {a_i} and {b_i} have to evolve.
1 (共1页)
进入Science版参与讨论
相关主题
Re: is it a joke or a surprise? MgB_2 is a high temperature superconduRe: 谁给俺讲讲Gauss-Chebyshev quadrature formula
Re: 问个最小二乘法的问题Re: 请教积分,急!
Re: 一个有趣的数学问题Re: how to compute the expected value in Mathematica?
Re: 求解两道数学题!帮我想想这个数学展开吧。
Re: 数列问题Re: Summation of Sequence
淋雨的概率problem: travelling wave in a ring network
Re: [转载] 高手们帮个忙Re: A set problem
Re: what is 经典力学中的Hamilton方程?help with numerical integral transformation
相关话题的讨论汇总
话题: lim话题: infty话题: sum话题: 无穷小