I want to verify if the integral $\int_1^{\infty}\ln((\exp(1/x)+(n-1))/n)dx, \;\;n>0$
is convergent or divergent? I made some handy calculations but couldn't find any way for it. Please give me help.
I want to verify if the integral $\int_1^{\infty}\ln((\exp(1/x)+(n-1))/n)dx, \;\;n>0$
is convergent or divergent? I made some handy calculations but couldn't find any way for it. Please give me help.
Note that $e^{1/x}+(n-1)/n\geq e^{1/x}$ and so $\ln (e^{1/x}+(n-1)/n)\geq 1/x$. Hence the given integral is greater than or equal to $\int_1^\infty\frac{1}{x}dx=\lim_{x\to\infty}\ln x=\infty$
HINT: Besides to @pritman's answer; consider the integrand as $f(x)=\ln(1+\frac{e^{x^{-1}}-1}{n})$. I agree with @did that $f(x)=\ln(1+\frac{e^{x^{-1}}-1}{n})\sim\frac{1}{nx}$ when $n$ tends to infinity. Can you do the rest?