limn→∞1nlog(1+1n)
I start by writing limn→∞1n=0, and limn→∞1n+log(11n)=0
Since limit exists, by multiplication law, limn→∞1nlog(1+1n)=0
Hence the series ∑∞n1nlog1+1n converge.
However, the limn→∞1nlog(1+1n) is given to be 1 in the solution, would really like to know where I've done wrong.
Thanks for the help.
Answer
Your first limit limn→∞1n+log(1+1n)=0
In the corrected case we have limn→∞1log(1+1n)n=1 since limn→∞log(1+1n)n=log(e)=1 if log means logarithmus naturalis, to the base e
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