Time complexity in Big O notation for Harmonic series with first k terms missing
We can estimate your sum using an integral:
$$ \int_k^{n+1} \frac{dx}{x} \leq \sum_{i=k}^n \frac{1}{i} \leq \int_{k-1}^n \frac{dx}{x}. $$
Evaluating the integrals, we obtain
$$ \log \frac{n+1}{k} \leq \sum_{i=k}^n \frac{1}{i} \leq \log \frac{n}{k-1}. $$
We can conclude that for $k \geq 2$,
$$ \sum_{i=k}^n \frac{1}{i} = \log \frac{n}{k} \pm O(1). $$