SL 1.8—Sum of infinite geo sequence: IB DP Math AA Notes

What is an Infinite Geometric Series?

A geometric sequence normally has a fixed number of terms — but what happens if we add up infinitely many terms?

Infinite Geometric Sequence

A geometric sequence that continues indefinitely, with no final term. Each term is obtained by multiplying the previous term by a constant common ratio $r$ . Example: $1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \dots$

Infinite Geometric Series

The sum of all terms of an infinite geometric sequence. Written as $u_{1} + u_{1} r + u_{1} r^{2} + u_{1} r^{3} + \dots$ This is what we denote $S_{\infty}$ , and it may or may not equal a finite number.

It is important to distinguish between the two:

The sequence is the list of terms: $u_{1}, u_{1} r, u_{1} r^{2}, \dots$
The series is the sum of those terms: $u_{1} + u_{1} r + u_{1} r^{2} + \dots$

The key question is: does the series (the running total) blow up to infinity, or does it settle towards a finite value?

If the terms grow larger and larger, the series has no finite value — it diverges.
If the terms shrink towards zero, the partial sums may approach a fixed number — the series converges.

Example

Consider an infinite geometric series with $u_{1} = 1$ and $r = \frac{1}{2}$ .

The terms are: $1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \dots$

The partial sums are: $1, 1.5, 1.75, 1.875, 1.9375, \dots$

Each term is half the previous one — the terms are getting closer and closer to 0, and the partial sums are getting closer and closer to 2. This series is convergent, and $S_{\infty} = 2$ .

Analogy

Imagine walking towards a wall that is 2 m away, but each step you take is exactly half the distance remaining. You step 1 m, then 0.5 m, then 0.25 m... The total distance you have walked approaches 2 m as a limit — that limiting total is the "sum to infinity". The wall at 2 m represents that limit value $S_{\infty}$ .

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What is an Infinite Geometric Series?

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