Find fast exact value for numbers in the form $\sum_{k=min}^{Max}\frac{1}{k}$

Question

I know I could start multiplying by all denominators and try to get the exact value that way but is there some smarter way or shortcut?

Let's take simple example: $\displaystyle \frac{1}{99}+\frac{1}{98}+...+1$. How to approximate or to get the exact value fast?

I know I could split the sequence into sum of geometric series like $$s_{2}=\frac{1}{2}+\frac{1}{4}+...=2,\qquad s_{3}=\frac{1}{3}+\frac{1}{9}+...=\frac{3}{2},$$ but there can be an infinite amount of them if $Max$ is infinite.

Answer 1

You can use the formula $\sum_{k=l+1}^h 1/k = H_h - H_l$ together with estimates for the harmonic number (here $H_t$ is the $t$th harmonic number). This gives for example the estimate $$\sum_{k=l+1}^h = \ln \frac{h}{l} + O\left(\frac{1}{l}\right).$$