On page 72 of the book Markov Chains by Norris, they state that, for $0<\lambda_n<\infty \;\; \forall n$:
$\sum_{n=1}^\infty 1/ \lambda_n = \infty$
implies
$ \prod_{n=1}^\infty (1+1/\lambda_n)=\infty$.
How can this be seen? I tried taking logarithms but that does not seem to help.
Is the reverse statement also true?