### Integral Test for Convergence

Posted:

**Tue Oct 21, 2014 8:37 am**$$\sum_1^\infty \ln(1+1/n)$$ is divergent.

I understand what the solution is saying (multiplying and eliminating everything except end terms....), but I couldn't figure out what I did wrong using the integral test: f(x) = ln (1+1/x) is decreasing, and positive, so $$\int_1^\infty \ln(1+1/x)dx = \int_1^\infty \ln(x+1)- lnx dx$$. Then letting y = x + 1 yields $$\int_2^\infty lnydy- \int_1^\infty lnx dx$$. It is integrating the same function from different starting points, won't the difference be a finite area?

Can anyone explain? Thanks!

I understand what the solution is saying (multiplying and eliminating everything except end terms....), but I couldn't figure out what I did wrong using the integral test: f(x) = ln (1+1/x) is decreasing, and positive, so $$\int_1^\infty \ln(1+1/x)dx = \int_1^\infty \ln(x+1)- lnx dx$$. Then letting y = x + 1 yields $$\int_2^\infty lnydy- \int_1^\infty lnx dx$$. It is integrating the same function from different starting points, won't the difference be a finite area?

Can anyone explain? Thanks!