Hey hedi.
Hint: Look at the taylor series expanded around a = 1 (i.e. taylor series for log(1+x)).
Let be the -th Harmonic number (the sum of the first terms of the Harmonic series). Consider partial sums of three terms of your series at a time. Show that gives the correct partial sums for your series. So, your series is given by .
Next, use common inequalities for the Harmonic numbers. For example, Young (1991) showed that . So,
and
Adding these inequalities together gives:
Simplifying gives:
Finally, use the Squeeze Theorem to show that , so
I'm curious what course (if any) this would be learned in. I've had a fair bit of math and never ran into a detailed analysis of the Harmonic series like this.
Is this something you just naturally come across post-grad in the course of solving other problems?