# Math Help - Proving converging and diverging

1. ## Proving converging and diverging

Show that if an > 0 and Σan is convergent, then Σln(1+an ) is convergent.

2. limit comparison test with $a_n,$ then let's compute $\lim_{n\to\infty}\frac{\ln\big(1+a_n\big)}{a_n},$ substitute $t=a_n$ then $t\to0$ since $a_n\to0$ because of the convergence of the first series, thus $\lim_{t\to0}\frac{\ln(1+t)}t=1$ and the second series converges.

or for each $n\ge1$ we have $\ln(1+n)\le n$ then $\ln(1+a_n)\le a_n$ and the rest follows by direct comparison test. (Comparison test does apply since $a_n>0,$ so our work is okay.)