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I am interested in the following questions:

given:

$$G(n) = \left(\frac12 + \frac23 + \frac34 + \frac45 + \cdots + \frac{n-1}n\right)n$$

  1. what is a $F(n)$ which could be an upper bound (clearly as tight as possible) for $G(n)$ for $n$ arbitrarily large ?

  2. Does the series: $$\frac12 + \frac23 + \frac34 + \frac45 + \cdots + \frac{n-1}n$$ have a "name" and a sum (any reference)?

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    This question most likely does not belong here, but should be migrated to math.SE. This site is specifically about software _Mathematica_, even though _Mathematica_ can do symbolic mathematics. If you have access to _Mathematica_, you can just type your query in and it will provide you with the answer. If you do not have access to [_Mathematica_](http://www.wolfram.com/mathematica), you could try using [Wolfram|Alpha](http://www.wolframalpha.com) which is built using _Mathematica_ and is quite capable of evaluating such a simple sum as yours.2012-11-17

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