An infinite geometric sequence is a geometric sequence that continues indefinitely. One is considered convergent if its terms approach a finite limit as the number of terms increases indefinitely.
For example, let's consider an infinite geometric sequence with $u_1 = 1$ and $r = \frac12$. The terms of this series go: $1, \frac12, \frac14, \frac18, \frac1{16}...$. Eventually, as $n$ goes to infinity, each term gets closer and closer to $0$, so we say it is convergent.
Whether an infinite geometric sequence converges depends on the value of its common ratio.
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