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By the end of these notes, you will be able to:
You already know that a geometric progression (GP) is a sequence where each term is found by multiplying the previous term by a fixed number called the common ratio, r.
An infinite geometric series is what you get when you add up the terms of a GP that goes on forever — it never stops.
For example: 2+1+21+41+81+⋯
Here, the first term a = 2 and the common ratio r = ½. The three dots (…) mean the series continues forever.
This is the big question. If you keep adding more and more terms, does the total eventually "settle" at a fixed number, or does it keep growing without limit?
Let's look at the example above and see what happens to the partial sums (the sum of the first n terms):
| Number of terms (n) | Sum so far (Sₙ) |
|---|---|
| 1 | 2 |
| 2 | 3 |
| 3 | 3.5 |
| 4 | 3.75 |
| 5 | 3.875 |
| … | … getting closer and closer to 4 |
Notice that as you add more and more terms, the total creeps closer and closer to 4, but never exceeds it. The total is converging (settling) on the value 4.
This means the sum to infinity of this series is 4.
💡 Think of it this way: Imagine a 2 × 2 square (total area = 4). You shade half of it, then half of what remains, then half of that, and so on. The shaded area keeps growing, but it can never exceed the total area of the square — 4. This is exactly what the series above is doing.
A series is called convergent when the sum of its terms approaches a specific finite number as you add more and more terms. It "homes in" on one value and stays there.
A series is called divergent when the sum keeps growing (or fluctuating wildly) without ever settling on a fixed number. There is no sum to infinity for a divergent series.
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