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I am trying to remember what the series 1+2+3+4+5+...+n is equal to in order to determine the series of breaks within the graph of x[x]. I know it obviously diverges as it goes to infinity, but what is the equation for when n is finite?

The sequence for the series goes 1,3,6,10,15,21,28,36,45,55,66,78,...

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marked as duplicate by callculus, user147263, Jyrki Lahtonen Nov 22 '15 at 6:50

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  • $\begingroup$ No, that question is asking for a name, not an equation. $\endgroup$ – The Great Duck Nov 22 '15 at 4:51
  • $\begingroup$ The formula is mentioned in several answers. Anyway there are many other similar questions. This question is probably the question with the most duplicates. $\endgroup$ – callculus Nov 22 '15 at 4:53
  • $\begingroup$ No Normal Human, my question is not math.stackexchange.com/questions/78936/…. That question is asking about discrete mathematics and an alternating power series. $\endgroup$ – The Great Duck Nov 22 '15 at 5:16
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The general term of the series you are considering is $$a_n=\sum_{i=1}^n i=\frac{1}{2} n (n+1)$$ as you can see here.

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Let $S_n=1+2+3+\cdots+n=n+(n-1)+(n-2)+\cdots+1.$. Then, $$2S_n=(n+1)+(n+1)+(n+1)+\cdots+(n+1)=n(n+1).$$

Hence, $$S_n=\frac{n(n+1)}{2}.$$

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