I have to find the following limit:
limn→∞n∑k=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)≤n∑k=0(nk)1n2n+k≤2nn2n.
By squeezing,
limn→∞n∑k=0(nk)1n2n+k=0.
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