CAIE IGCSE Additional Math 12.5 convergence and sum to infinity

Use this convergence page to test whether |r|<1, justify whether a sum to infinity exists and solve related GP equations.

Exam points

Explain whether a geometric progression has a sum to infinity by checking |r|<1.
Use the sum-to-infinity formula after finding the first term and common ratio.
Solve equations from GP terms before deciding which value gives convergence.

[Maximum number: 5]

The first three terms of a different geometric progression are $\frac{4}{3} \cos ^{2} 3 \theta, \frac{16}{9} \cos ^{4} 3 \theta$ and $\frac{64}{27} \cos ^{6} 3 \theta$ ,
for $0<\theta<\frac{\pi}{3}$ . Find the set of values of $\theta$ for which this progression has a sum to infinity.

9(c) Common ratio $=\frac{4}{3}\cos^{2}3\theta$ B1
Allow unsimplified. Must be convinced it is the common ratio, not just writing the first term e.g. $r=$ , or seeing
$\frac{\frac{16}{9}\cos^{4}3\theta}{\frac{4}{3}\cos^{2}3\theta}$
For a sum to infinity, $|r|<1$ , so
$\begin{aligned} -1 &< \frac{4}{3}\cos^{2}3\theta < 1,\\ 0 &\leqslant \frac{4}{3}\cos^{2}3\theta < 1,\\ -\frac{\sqrt{3}}{2} &< \cos 3\theta < \frac{\sqrt{3}}{2}. \end{aligned}$
B1 for the valid ratio condition; B1 for the corresponding cosine bounds.
$\frac{\pi}{6}<3\theta\leqslant\frac{5\pi}{6}$ B1
$\frac{\pi}{18}<\theta\leqslant\frac{5\pi}{18}$ B1

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12.5. Use convergence and sum to infinity question 1