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Geometric Series one would use $S_n = \dfrac{a_1\cdot (1 - r^n)}{(1 -r)}$. Arithmetic Series one would use $S_n = \dfrac{n\cdot (a_1 + a_n)}{2}$.
But how would I convert a sigma notation problem with factorials in it to an equation? For example $$ \sum_1^n \frac{i}{ (i + 1)!} $$

How do I convert this into an equation. (In particular I am trying to create another equation and use induction to prove that the two equations equal each other but that is just the application of why I need to find another way of writing this formula and if it was dealing with Geometric Series or Arithmetic Series I could do it but with factorials I am stuck.)

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The trick is to write $\dfrac{i}{(i+1)!} = \dfrac{1}{i!} - \dfrac{1}{(i+1)!}$ and use telescope. So the use of either geometric or arithmetic sequence formulas is not necessary.

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  • $\begingroup$ Whats a good link to read on "telescope" or what should I search for in order to find the meaning of it? Second, I see that they equal each other and yes what is the trick? How did you do this? $\endgroup$
    – Adam
    Apr 14 '14 at 6:34
  • $\begingroup$ Try en.wikipedia.org/wiki/Telescoping_series $\endgroup$ Apr 14 '14 at 7:21

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