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Show, rigorously, that the geometric series $a + ar +ar^2 + \cdots + ar^{n-1} + \cdots$ converges if and only if |r| < 1.

Also, show that if |r| < 1, the sum is given by $S = \frac{a}{1-r}$.


Note(s):

Question 1, First part: if |r| < 1, show (rigorously) that the geometric series above converges.

Update: I'm reposting this question as the accepted answer on the possible duplicate question doesn't provide a rigorous [$\epsilon$-$\delta$] derivation of an important step, namely, $\lim_{N\to\infty}r^{N+1} = 0$ if $|r|\lt 1$. [Translated to the current question's context, $\lim_{n\to\infty}ar^n = 0$ if $|r|\lt 1$].

Update 2: The problem of the above issue has been addressed.


Question 1, Second part: if the geometric series above converges, then show (rigorously) that |r| < 1.

-- Any hints would be appreciated. --

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    @UGPhysics: In any case, I have added$a$proof of this limit in [the original answer](http://math.stackexchange.com/questions/29023/value-of-sum-xn/29035#29035). It uses the infimum property, the Archimdean property, and that$a$subsequence of a converging sequence must converge to the same limit as the original sequence. The latter is very easy; the previous two would require a formal construction of the real numbers in order to provide a formal proof.2011-09-30

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