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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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    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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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 $\sum_{k=l+1}^h = \ln \frac{h}{l} + O\left(\frac{1}{l}\right).$

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    @hhh, but I have no idea what $d/dn(H_n)=(1/6)(\pi^2-6H_n^2)$ means, because I don't understand what the left side means. How are you defining the derivative of $H_n$?2011-06-03