Test convergence and find a sum to infinity
Decide whether a geometric progression converges and calculate its infinite sum only when the common ratio satisfies the convergence condition.
Section 12 · Series
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What condition gives a finite sum to infinity for a GP?
What you will learn
- Use |r| < 1 as the condition for a geometric progression to converge
- Find the sum to infinity of a convergent progression
- Explain why a progression with |r| ≥ 1 has no finite sum to infinity
Terms must approach zero for the infinite sum to settle
A geometric progression with first term a and ratio r has a finite sum to infinity only when |r| < 1. Then each later term shrinks towards zero and the partial sums approach a limit.
If r ≥ 1 or r ≤ −1, the terms do not approach zero, so the partial sums do not settle to a finite limit. A geometric-looking sequence does not always have a sum to infinity.
Use the infinite-sum formula only after checking convergence
For |r| < 1, S∞ = a/(1 − r). This formula is provided in the 0606 formula list. A negative ratio is allowed if its magnitude is less than 1; the terms then alternate while shrinking.
State the convergence check before calculating. For example, 5 − 2.5 + 1.25 − … has r = −1/2, so it converges and its sum is 5/[1 − (−1/2)] = 10/3.
Pause and explain
Can the series 3 − 6 + 12 − 24 + … have a finite sum to infinity?
Worked example
Find the sum to infinity of 2 + 1 + 1/2 + 1/4 + …
Show the worked solution
- The first term is 2 and the ratio is 1/2.
- Since |1/2| < 1, the progression converges.
- S∞ = 2/[1 − 1/2] = 4.
Answer The sum to infinity is 4.
Try a topic test
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Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find the sum to infinity of 5 − 2.5 + 1.25 − …
Give me a hint
Check the signed ratio and its magnitude before applying the formula.
Compare my reasoning
- a = 5 and r = −1/2.
- Since |r| = 1/2 < 1, the series converges.
- S∞ = 5/[1 − (−1/2)] = 10/3.
10/3.
Look for these in your work
- Ratio sign preserved
- Convergence condition stated
- Infinite sum calculated
Explain whether 3, 6, 12, 24, … has a sum to infinity.
Give me a hint
Compare the common ratio's magnitude with 1.
Compare my reasoning
- The ratio is 2.
- Its magnitude is not less than 1, and the terms grow rather than tend to zero.
- Therefore the partial sums cannot approach a finite value.
It does not converge and has no finite sum to infinity because |2| ≥ 1.
Look for these in your work
- Common ratio identified
- Convergence condition applied
- Reason explained
A bouncing ball travels 50 cm on its first measured rise; each later rise is 20% of the previous one. Find the total rise distance over all bounces, assuming this geometric model continues.
Give me a hint
Use a = 50 and r = 0.2; verify convergence first.
Compare my reasoning
- The ratio has magnitude 0.2 < 1, so the series converges.
- S∞ = 50/(1 − 0.2).
- Calculate 50/0.8.
The total rise distance is 62.5 cm.
Look for these in your work
- Ratio model interpreted
- Convergence checked
- Units retained
Common mistakes
- Using the formula when |r| ≥ 1
- Checking r < 1 but forgetting that a negative r can have |r| greater than 1
- Using 1 + r instead of 1 − r in the denominator
- Assuming finite terms guarantee that an infinite sum exists
What can you explain now?
Does a GP with ratio −0.8 converge?
Compare with the explanation
Yes
Its ratio has magnitude 0.8, which is less than 1.
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Make it stick
What condition must hold before using S∞ = a/(1 − r), and why?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.