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Thursday, January 4, 2018

Exercise from discuss.com.hk (2)

Problem:

limnn1k=1k2k(nk)

Solution:

Let p=nk, so we have

n1k=1k2k(nk)=n1p=1(np)2npp=12nn1p=1(np)2pp=12nn1p=1(np)2pp=12nn1p=1(n2ppp2pp)=12nn1p=1(n2pp2p)=n2nn1p=12pp12nn1p=12p=n2nn1p=12pp12n(2n1)=n2nn1p=12pp1+12n

Notice the first term look very much like the previous post, so we will use the same trick to solve the problem, let L(n)=n2nn1p=12pp

L(n+1)=n+12n+1np=12pp=n+12n+1(n1p=12pp+2nn)=n+12n+1n1p=12pp+n+12n=n+12n+12nnn2nn1p=12pp+n+12n=n+12n+12nnL(n)+n+12n=n+12nL(n)+n+12n

Again, assume the limit exists, taking limits on both sides, we get

L=12L+122L=L+1L=1

Therefore the overall solution is simply 1 - 1 = 0.

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