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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.

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    If Max is infinite then the sum diverges.2011-06-02
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    Your "simple example" isn't an example, as min is 1 and max is 99 but everything between 1 and 98 is missing. What do you really mean?2011-06-02
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    @Gerry Myerson: sorry forgot the dots, corrected.2011-06-02
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    OK, then Yuval has given you a good answer. $\sum_1^n(1/k)=\log n+\gamma+$ terms of lower order, where $\gamma=.57721\dots$ is the Euler-Mascheroni constant. A websearch for gamma, or Euler-Mascheroni, or harmonic number, will get you tons of information.2011-06-02

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