I am trying to check the convergence or divergence of the series ∞∑n=11nlog(1+1n).
My attempt: for a finite
p,n+p∑k=n1klog(1+1k)<1nn+p∑k=nlog(1+1k)=1nlogΠn+pk=n(k+1k)=1nlog(1+p+1n)<1nlog2, for large n and p is finite.<ε
Hence the series converges.
Answer
Because 1nln(1+1n)1n2→1 and
∞∑n=11n2=π26.
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