What is an Infinite Geometric Sequence?
A geometric sequence is one where each term is obtained by multiplying the previous term by a fixed value called the common ratio, . Most of the time in IB Maths, we deal with finite sequences — but sequences can also continue indefinitely.
A geometric sequence that has no final term — it continues forever. The general term is for all positive integers .
The key question with any infinite sequence is: what happens to the terms as ? Do they grow without bound, or do they settle toward zero?
Consider the sequence with and :
Each term is exactly half the previous one. The terms get closer and closer to zero but never actually reach it. This sequence is said to converge.
Now consider and :
The terms grow without bound. This sequence diverges — there is no finite limit.