An infinite geometric series is the sum of an infinite geometric sequence. This series would have no last term. The general form of the infinite geometric series is $a_{1}+a_{1}r+a_{1}r^{2}+a_{1}r^{3}+….$ ,where $a_{1}$r is the first term and r is the common ratio.
The Infinite Geometric Series Formula is given as,
\[\large a_{1}+a_{1}r+a_{1}r^{2}+a_{1}r^{3}+….+a_{1}r^{n-1}\]
The formula for the resultant sum of the Infinite Geometric Series is,
\[\large S_{\infty}=\frac{a_{1}}{1-r};\left|r\right|<1\]
The following table shows several geometric series with different common ratios:
| Common ratio, r | Start term, a | Example series |
|---|---|---|
| 10 | 4 | 4 + 40 + 400 + 4000 + 40,000 + ··· |
| 1/3 | 9 | 9 + 3 + 1 + 1/3 + 1/9 + ··· |
| 1/10 | 7 | 7 + 0.7 + 0.07 + 0.007 + 0.0007 + ··· |
| 1 | 3 | 3 + 3 + 3 + 3 + 3 + ··· |
| −1/2 | 1 | 1 − 1/2 + 1/4 − 1/8 + 1/16 − 1/32 + ··· |
| –1 | 3 | 3 − 3 + 3 − 3 + 3 − ··· |
Solved Examples
Question: Find the sum of the geometric series 125 + 25 + 5 + 1 +…… ?
Solution:
The series is, 125 + 25 + 5 + 1 +…..
a1 = 125
r = $\frac{25}{125}$ = $\frac{1}{5}$
The formula for the resultant sum of the Infinite Geometric Series is,
S∞ = $\frac{a_{1}}{1 – r}$
S∞ = $\frac{125}{1 – \frac{1}{5}}$
S∞ = $\frac{125}{\frac{4}{5}}$
S∞ = $\frac{625}{4}$