Tuesday, 22 April 2014

calculus - Determine limntoinftynleft(1+(n+1)lnfracnn+1right)



Determine lim
I noticed the indeterminate case \infty \cdot 0 and I tried to get them all under the \ln, but it got more complicated and I reached another indeterminate form. The same happened when I tried to use Stolz-Cesaro.



EDIT: is there an elementary solution, without l'Hospital or Taylor series?



Answer



Let \frac {n}{n+1}=e^x. As n\to \infty,\;\; x\to 0 through negative values.The expression is e^x\cdot \frac {1+x-e^x}{(1-e^x)^2}. Applying l'Hopital's Rule to \frac {1+x-e^x}{(1-e^x)^2} we get a limit of \frac {-1}{2} as x\to 0. And the far left term e^x in the expression goes to 1 as x\to 0.


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