The sum of a geometric series
One may be curious to find a sum of a geometric series &sumqk. Does any such exist? That is, is it possible to associate a number with the expression
The answer depends on q. For
The only way to define such a sum is by appealing to the theory of limits. By definition,
where ∑n is the partial sum of all the terms from the first and up to the nth which is qn.
The latter can be easily evaluated. Since
q ∑n is found by multiplying that term-by-term by q:
Note that the two sums share all the terms except for the first (1) that appears only in the first sum ∑n and the term
We wish to define ∑ as limn→∞∑n, i.e.
The latter exists and equals 1 / (1 - q) iff |q| < 1, for then and only then limn→∞qn +1 exists and when it does it equals 0.
For a more general series that starts with an arbitrary term a,
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