Question

Show that if a_n >0 and summation of a_n is convergent, then summation of ln(1+a_n) is convergent

Answer 1

Use the majorizing criteria.

Answer 2

If sum of a_n converges, then lim a_n =0.(as n goes to infinity)
If lim a_n =0, then lim ln(1+a_n)=0.
If lim ln(1+a_n)=0, then sum of ln(1+a_n) converges.

Answer 3

That is not true. Just knowing that the elements of the series converge to 0 does not mean that the series converges. Take 1/n for example where the series diverges.

Answer 4

ok thanks i wasn't sure if my reasoning was correct there.
What would be the correct reasoning for showing this?
you could show that ln(1+a_n) <= a_n for all n>=0

Answer 5

yes, that is true.

Answer 6

thanks

Answer 7

sevenlash, did you get this problem sorted? If not, let me know. I looked at it and have a solution.

Answer 8

Just not much point writing it out unless you need it :)