Grade 12 Maths · Number Patterns, Sequences and Series

Use the convergence condition

Decide if a geometric series adds up to a fixed number as it goes on forever, which happens only when −1 < r < 1.

past paper questions use it: 11 past paper questions use it
videos: 4 videos

Learn it: use the convergence condition

Pick one and watch it, then try the questions below. Videos load only when you tap play.

11) Converging series Gr 12 | Intro Part 1Kevinmathscience

11a) For which x value will converge | TryKevinmathscience

11) Converging series Gr 12 | Intro Part 3Kevinmathscience

Value of x for which series converges for Grade 12Academia e-learning

Now try it on a real exam question

No sign-up. Takes 3 minutes.

Question 2.1.1 · May/June 2024 · Paper 1 · National (DBE) · 2 marks

2.1 Consider the geometric series:

4 + 2 + 1 + \frac{1}{2} + \dots

2.1.1 Does the series converge? Justify your answer.(2)

Work it out on paper first. Then mark yourself with the official memo, one tick at a time.

More past paper questions: use the convergence condition

Work each one out first, then open the memo and check every tick.

3. Question 3.3.2 · March 2021 · Provincial · 2 marks
3.3 Given: $\frac{24}{x} + 12 + 6 x + 3 x^{2} + \dots; x \neq = 0$
3.3.2 Write down the value of $x$ for which the series is increasing.(2)
Show the memo (1 tick)Memo
1. ✓
  Determine condition for increasing geometric series $r = \frac{x}{2}$ ; the terms grow when $\frac{x}{2} > 1$ | $x < - 2$ or $x > 2$
  2 marks · Use the convergence condition
4. Question 1.1.3 · March 2021 · Provincial · 2 marks
1.1 The following geometric sequence is given: $10; 5; 2, 5; 1, 25; ...$
1.1.3 Explain why the infinite series $10 + 5 + 2, 5 + 1, 25 + ...$ exists.(2)
Show the memo (1 tick)Memo
1. ✓
  Condition for convergence with r $- 1 < r < 1$ | $∴ - 1 < \frac{1}{2} < 1$
  2 marks · Use the convergence condition
5. Question 1.2.2 · March 2025 · Provincial · 2 marks
Consider the arithmetic sequence: $8; 15; 22; ...$
1.2 $(x - 2)^{2} + (x - 2)^{3} + (x - 2)^{4} + ...$ forms a geometric series.
1.2.2 Determine the value(s) of $x$ for which the series will converge.(2)
Show the memo (2 ticks)Memo
1. ✓
  Apply convergence condition $- 1 < r < 1$
  $- 1 < x - 2 < 1$
  1 mark · Use the convergence condition
2. ✓
  Solve for x $1 < x < 3$
  1 mark · Solve a linear inequality
6. Question 3.1.2 · March 2023 · Provincial · 2 marks
$(x - 2)^{2} + (x - 2)^{3} + (x - 2)^{4} + ...$ form a geometric series.
3.1 Geometric series properties
3.1.2 Determine the value(s) of $x$ for which the series will converge.(2)
Show the memo (2 ticks)Memo
1. ✓
  Apply convergence condition $- 1 < r < 1$
  $- 1 < x - 2 < 1$
  1 mark · Use the convergence condition
2. ✓
  Solve for x $1 < x < 3$
  1 mark · Solve a linear inequality

Free on Android

Know exactly which skill is costing you marks.

Mark past papers tick by tick in the NavyBlue app. It remembers every mark you lose, shows you the skills to fix first, and picks your next questions.

The NavyBlue app showing a learner's weak skills