I am interested in the asymptotic bounds of the following recurrence:
$$T(n)= T(n - 1) + \frac{1}{n\log n}$$
with base case $T(1) = 1$. I'm having trouble while solving this recurrence. It seems much more tricky than I initially thought.
I've tried out by iterative method and since $\log(n!) = n\log n$, I've got $\frac{1}{n\log n}$.
At this point, I've got that the summation of $\frac{1}{i}$ is equal to $\Theta(\log n + O(1))$. So at the end, I've come up with this bound (by simplifying the two terms): $\Theta(\frac{1}{n})$.
I'm not so sure about this bound. What is a good way to approach this problem?