Problem:
limn→∞n−1∑k=1k2k(n−k)
Solution:
Let p=n−k, so we have
n−1∑k=1k2k(n−k)=n−1∑p=1(n−p)2n−pp=12nn−1∑p=1(n−p)2−pp=12nn−1∑p=1(n−p)2pp=12nn−1∑p=1(n2pp−p2pp)=12nn−1∑p=1(n2pp−2p)=n2nn−1∑p=12pp−12nn−1∑p=12p=n2nn−1∑p=12pp−12n(2n−1)=n2nn−1∑p=12pp−1+12n
L=12L+122L=L+1L=1
limn→∞n−1∑k=1k2k(n−k)
Solution:
Let p=n−k, so we have
n−1∑k=1k2k(n−k)=n−1∑p=1(n−p)2n−pp=12nn−1∑p=1(n−p)2−pp=12nn−1∑p=1(n−p)2pp=12nn−1∑p=1(n2pp−p2pp)=12nn−1∑p=1(n2pp−2p)=n2nn−1∑p=12pp−12nn−1∑p=12p=n2nn−1∑p=12pp−12n(2n−1)=n2nn−1∑p=12pp−1+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)=n2nn−1∑p=12pp
L(n+1)=n+12n+1n∑p=12pp=n+12n+1(n−1∑p=12pp+2nn)=n+12n+1n−1∑p=12pp+n+12n=n+12n+12nnn2nn−1∑p=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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