Infinite Geometric Series | CAIE 9709 Pure Mathematics 1 Notes

Pure Mathematics 1

Mathematics9709

A sum without a last term

For $1+\tfrac12+\tfrac14+\cdots$ , the successive totals are 1, 1.5, 1.75, 1.875, …. These are : each adds only a finite number of terms.

They approach 2 as closely as we require by taking enough terms. We call 2 the , and say the series . The sum to infinity means this limit, not a last partial sum.

First term a = 1

Common ratio r0.5Number of terms n12

Converges to 2

Use the same scale for r and −r

Term uₙ: ≈ 0.0004883
Partial sum Sₙ: ≈ 2
Distance from limit: ≈ 0.0004883

Values are rounded to 4 significant figures. A displayed sum may round to the limit while a non-zero distance remains.

Position	Term	Partial sum
1	1	1
2	0.5	1.5
3	0.25	1.75
4	0.125	1.875

At r = −0.5, predict whether increasing n from 5 to 6 moves the sum closer to its limit. Choose an answer to show n = 6.

Green dashes mark the limit. Points represent integer n; dotted joins guide the eye. The vertical scale stays fixed as n changes.

Compare $r=0.5$ , $r=-0.5$ and $r=-1$ . Changing signs does not prevent convergence; failing to approach one value does.

Check the ratio before finding the sum

For a GP with non-zero first term, $|r|<1$ means $-1<r<1$ . Then $r^n$ approaches zero, so the finite-sum formula approaches

S_\infty=\frac{a}{1-r},\qquad |r|<1

The symbol $\infty$ means infinity. MF19 supplies this formula and its condition.

With $r=1$ , non-zero terms keep adding the same amount.
With $r=-1$ , the sums alternate between $a$ and zero. Neither endpoint is allowed.
With $|r|>1$ , the added terms grow in magnitude. There is no finite sum to infinity.

If $a=0$ , every term is zero and the sum is zero. The usual ratio condition assumes a non-zero first term. Also, “terms approach zero” alone does not prove convergence for a non-geometric series.

Worked example

Find the sum to infinity of $5+2+0.8+\cdots$ .

$r=2/5$ satisfies $|r|<1$ .

S_\infty=\frac5{1-2/5}=\frac{25}{3}

Now you try

Your turnIndependent practice

(a) Find the sum to infinity of $12-6+3-\cdots$ .

(b) A non-zero GP has ratio $2x-1$ . Find the values of $x$ for which it converges.

Show worked answer

(a) $r=-1/2$ , so $S_\infty=12/(1+1/2)=8$ .

(b) Solve $-1<2x-1<1$ : $0<x<1$ . Both ends are excluded.

(c) The ratio is positive, so require $2\cos\theta<1$ . Cosine decreases on this interval and equals $1/2$ at $\pi/3$ . Hence $\pi/3<\theta<\pi/2$ .