Wednesday, March 8, 2017

limits - Evaluate limlimitsntoinftyfrac1nlogleft(1+frac1nright)



limn1nlog(1+1n)



I start by writing limn1n=0, and limn1n+log(11n)=0



Since limit exists, by multiplication law, limn1nlog(1+1n)=0




Hence the series n1nlog1+1n converge.



However, the limn1nlog(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 limn1n+log(1+1n)=0


In the corrected case we have limn1log(1+1n)n=1
since limnlog(1+1n)n=log(e)=1
if log means logarithmus naturalis, to the base e


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