Consider
1+21+...+n1+n+11+...+2n1=(1+21+...+n1)+(n+11+...+2n1)Then
n+11+...+2n1=(1+21+31+...+n1+n+11+...+2n−11+2n1)−2(21+41+...+2n1)=1−21+31−41+...+2n−11−2n1=k=1∑2nk(−1)k+1Consider ln(1+x) for x∈(−1,x]
ln(1+x)=k=1∑∞k(−1)k+1xkThen for x=1
ln2=ln(1+1)=k=1∑∞k(−1)k+1(1)k=k=1∑∞k(−1)k+1We see that
n→∞liman=n→∞lim(n+11+...+2n1)=n→∞limk=1∑2nk(−1)k+1=k=1∑∞k(−1)k+1=ln2Therefore the sequence (an) is convergent.
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