Sum
The sum
of a geometric series is finite as long as the terms approach zero; as
the numbers near zero, they become insignificantly small, allowing a sum
to be calculated despite the series being infinite. The sum can be
computed using the self-similarity of the series.
Example
Consider the sum of the following geometric series:
This series has common ratio 2/3. If we multiply through by this
common ratio, then the initial 1 becomes a 2/3, the 2/3 becomes a 4/9,
and so on:
This new series is the same as the original, except that the first term is missing. Subtracting the new series (2/3)
s from the original series
s cancels every term in the original but the first:
A similar technique can be used to evaluate any self-similar expression.
Formula
For
, the sum of the first
n terms of a geometric series is:
where
a is the first term of the series, and
r is the common ratio. We can derive this formula as follows:
As
n goes to infinity, the absolute value of
r must be less than one for the series to converge. The sum then becomes
When
a = 1, this simplifies to:
the left-hand side being a geometric series with common ratio
r. We can derive this formula:
The general formula follows if we multiply through by
a.
This formula is only valid for convergent series (i.e., when the magnitude of
r is less than one). For example, the sum is undefined when
r = 10, even though the formula gives
s = −1/9.
This reasoning is also valid, with the same restrictions, for the complex case.
Proof of convergence
We can prove that the geometric series converges using the sum formula for a geometric progression:
Since (1 +
r +
r2 + ... +
rn)(1−
r) = 1−
rn+1 and
rn+1 → 0 for |
r | < 1.
Convergence of geometric series can also be demonstrated by rewriting the series as an equivalent telescoping series.
Consider the function g(K) = (r^(K+1))/(1-r). Note that: r = g(0) -
g(1), r^2 = g(1) - g(2), r^3 = g(2) - g(3), . . . Thus: S = r + r^2 +
r^3 + . . . = (g(0) - g(1)) + (g(1) - g(2)) + (g(2) - g(3)) + . . . If
|r|<1, then g(K) -> 0 as K -> infinity, and so S converges to
g(0) = r/(1-r).
Applications
Repeating decimals
Main article: Repeating decimal
A repeating decimal can be thought of as a geometric series whose common ratio is a power of 1/10. For example:
The formula for the sum of a geometric series can be used to convert the decimal to a fraction:
The formula works not only for a single repeating figure, but also for a repeating group of figures. For example:
Note that every series of repeating consecutive decimals can be conveniently simplified with the following:
That is, a repeating decimal with repeat length
n is equal to the quotient of the repeating part (as an integer) and
10n - 1.
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