Infinite Series
These QuickNotes give you the most important points to remember for the ESSLCE. They are based on the MoE Grade 12 Mathematics Textbook, Unit 1, Chapter 1.4.
~7 min read- An infinite series adds the terms of a sequence that never ends.
- A geometric series converges and has a finite sum only when the common ratio satisfies −1 < r < 1.
- When it converges, the sum to infinity is S∞ = G1 ÷ (1 − r), and this lets you turn a recurring decimal into a fraction.
Key Words
- Infinite series: an infinite series is the sum of the terms of a sequence that has no last term.
- Converge: a series converges when its partial sums get closer and closer to one fixed number.
- Diverge: a series diverges when its partial sums do not settle on a fixed number, because they grow without limit or keep jumping.
- Sum to infinity (S∞): the sum to infinity is the single number that a convergent series gets closer and closer to.
- Recurring decimal: a recurring decimal is a decimal with a block of digits that repeats forever.
What This Chapter Is About
- This chapter teaches you what happens when you add the terms of a sequence that never stops.
- You learn when such a sum settles on a finite value and how to use that idea to change recurring decimals into fractions.
What an Infinite Series Is
- An infinite series is the sum a1 + a2 + a3 + … that keeps going forever.
- To study it, you look at the partial sums S1, S2, S3, and watch where they are heading.
- If the partial sums settle on one number, the series converges; if they do not, the series diverges.
Convergence and Divergence
- A sequence converges when its terms get closer and closer to one fixed value as n grows.
- A sequence diverges when its terms grow toward positive or negative infinity, or keep jumping between values without settling.
- For a geometric sequence with ratio r, the terms shrink toward zero when −1 < r < 1, so that series converges.
- When the size of r is 1 or larger, the terms do not shrink, so the geometric series diverges.
Note on the textbook: One worked example says that (2/3)n “diverges” because it goes to zero. Going to zero is exactly what convergence means, so a geometric sequence with −1 < r < 1 in fact converges.
Figure 1.4: when −1 < r < 1, the partial sums level off at a finite value, so the series converges; otherwise they grow without limit and the series diverges.
Sum to Infinity of a Geometric Series
- When −1 < r < 1, the term rn shrinks to zero as n grows, so the partial sums settle on one value.
- That value is the sum to infinity, and it gives a finite answer even though you are adding forever.
- If the size of r is 1 or larger, there is no sum to infinity, because the series diverges.
S∞ = G1 ÷ (1 − r), for −1 < r < 1
This means you divide the first term by (1 minus the common ratio). You may only use this when r lies strictly between −1 and 1.
Worked example: find the sum of 1 + 2/3 + 4/9 + …
Here G1 = 1 and r = 2/3, and the size of r is less than 1. So S∞ = 1 ÷ (1 − 2/3) = 1 ÷ (1/3) = 3.
Recurring Decimals as Fractions
- A recurring decimal can be written as an infinite geometric series, so you can find its exact fraction.
- You break the decimal into the repeating blocks, find G1 and r, then use the sum to infinity.
- A quick rule for a pure recurring decimal is to put the repeating digits over as many 9s as there are repeating digits.
Worked example: write 0.474747… as a fraction.
The block “47” repeats, so the fraction is 47 over 99, which gives 0.474747… = 47/99.
Quick Comparison Table
| Common ratio r | What the terms do | The series |
|---|---|---|
| −1 < r < 1 | Shrink toward zero | Converges, sum = G1 ÷ (1 − r) |
| r = 1 | Stay the same size | Diverges to infinity |
| r > 1 | Grow larger and larger | Diverges to infinity |
| r ≤ −1 | Keep jumping in sign | Diverges and vibrates |
Common Mistakes to Avoid
- Do not use the sum to infinity unless the size of r is less than 1, because otherwise no finite sum exists.
- Do not think a series diverges just because its terms go to zero, since terms shrinking to zero is what convergence looks like.
- Do not mix up the finite-sum formula with the infinite one, because S∞ drops the rn part once it shrinks away.
- Do not write the wrong number of 9s for a recurring decimal, because you need one 9 for each repeating digit.
Easy Ways to Remember
- A fraction r between −1 and 1 makes the terms fade away, so the endless sum can still be finite.
- For the sum to infinity, just remember “first term over one minus r”.
- For a pure recurring decimal, write the repeating block over the same number of 9s, so 0.7 repeating is 7/9.
Quiz
Tap an answer to check it.
1. A geometric series converges only when the common ratio r satisfies which condition?
Only when r lies strictly between −1 and 1 do the terms shrink to zero, so the series converges.
2. Find the sum to infinity of 8 + 4 + 2 + 1 + …
Here G1 = 8 and r = 1/2, so S∞ = 8 ÷ (1 − 1/2) = 8 ÷ (1/2) = 16.
3. Which series diverges?
The series 3 + 9 + 27 + … has r = 3, which is larger than 1, so its partial sums grow without limit and it diverges.
4. Write the recurring decimal 0.333… as a fraction.
One digit repeats, so you put 3 over a single 9, which gives 3/9 = 1/3.
5. Why can an endless sum still give a finite value?
When the terms shrink toward zero, each new term adds almost nothing, so the partial sums settle on one finite value.
Remember: An infinite geometric series converges only when −1 < r < 1, and then its sum to infinity is S∞ = G1 ÷ (1 − r). Terms that shrink toward zero are a sign of convergence, not divergence, and this idea turns any recurring decimal into a fraction.