An infinite geometric sequence is geometric sequence that continues indefinitely.
One is considered convergent if its terms approach a finite limit as the number of terms increases indefinitely.
Example
Let's consider an infinite geometric sequence with $u_1 = 1$ and $r = \frac12$.
The terms of this series go: $1, \frac12, \frac14, \frac18, \frac16...$.
Eventually, as the terms go to infinity, each term gets closer and closer to 0, so we say it is convergent.
Condition for Convergence
Whether an infinite geometric sequence converges depends on the value of its common ratio.
The condition for convergence is: $$|r|< 1$$ where $|r|$ represents the absolute value (or modulus) of the common ratio.
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Infinite Geometric Series
An infinite geometric series is a geometric series that continues indefinitely without stopping.
DefinitionconvergentWhen the absolute value of the common ratio is less than 1 (∣r∣<1), the series is called convergent, meaning it approaches a finite sum as more terms are added.
DefinitiondivergentIf ∣r∣≥1, the series is divergent, meaning the sum grows without bound or oscillates indefinitely.
ExampleConsider the infinite geometric series 1+21+41+81+…. Here, a=1 and r=21. Since ∣r∣<1, the series is convergent.
ExampleFor the series 5+10+20+40+…, we have a=5 and r=2. Since ∣r∣=2>1, the series is divergent.
