Sum to Infinity of a Geometric Progression

Sum to Infinity of a GP

When a GP goes on forever ($n \to \infty$) and the common ratio satisfies $-1 < r < 1$ (i.e. $|r| < 1$), the successive terms shrink toward zero and the running sum approaches a fixed, finite value called the sum to infinity, $S_\infty$:

If $|r| \geq 1$, the terms do not shrink (they stay the same size or grow), so the sum never settles down to a finite value — $S_\infty$ does not exist in that case.

Worked idea: for the GP $8, 4, 2, 1, \ldots$, $a = 8$ and $r = \tfrac{1}{2}$, so $S_\infty = \dfrac{8}{1 - \tfrac12} = \dfrac{8}{\tfrac12} = 16$.

A Practical Use: Converting a Recurring Decimal to a Fraction

A repeating decimal like $0.\overline{7} = 0.7777\ldots$ can be written as a GP: $0.7 + 0.07 + 0.007 + \cdots$, where $a = 0.7$ and $r = 0.1$ (each term is one-tenth of the one before). Since $|r| < 1$, its sum to infinity exists:

$S_\infty = \frac{0.7}{1 - 0.1} = \frac{0.7}{0.9} = \frac{7}{9}$ …