Sunday, 2 August 2015

calculus - Find limlimitsntoinftysumlimitsnk=0dfracbinomnkn2n+k.




I have to find the following limit:



limnnk=0(nk)n2n+k



I thought I can use something from this other, seemingly similar question, but I don't see any way of manipulating this sum into something easier to work with. So how should I approach this limit?


Answer



When k{0,1,,n}
(nk)1n2n+n(nk)1n2n+k(nk)1n2n,


whence
2nn(2n+1)nk=0(nk)1n2n+k2nn2n.




By squeezing,
limnnk=0(nk)1n2n+k=0.


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