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Is writing a sequence as a telescoping sum the only way to turn a sequence into an infinite series? In particular:

Let $s_n = 1 +\frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}- \ln(n+1), \ n \geq 1$. Convert this into an infinite series.

So let $a_n = s_n-s_{n-1}$. Then $a_n = \frac{1}{n}-\ln(n+1)+ \ln n$. This is equivalent to $$\sum_{n=1}^{\infty} \left(\frac{1}{n}- \ln \frac{n+1}{n} \right)$$

What does the inside term represent?

  • 0
    What do you mean what does it represent? There's no new notation or terminology there, and I assume you know what a logarithm is. Maybe have a look at http://en.wikipedia.org/wiki/Telescoping_series2011-07-04
  • 0
    You don't just need to take a limit, hmm?2011-07-04
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    I am afraid I do not even know what the question is.2011-07-04

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