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3.5PureStretch

Sum to Infinity

If a geometric series shrinks fast enough, its infinitely many terms add to a finite total. This happens exactly when the common ratio satisfies |r| < 1 — such a series is called convergent.

25 min Sequences and Series
Edexcel A level Maths: 3.5 Sum to InfinityCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Know when a geometric series converges
  • Use the sum to infinity formula
  • Distinguish convergent from divergent series
  • Solve problems involving S∞
1

Convergence condition

A geometric series converges (has a finite sum to infinity) only if ∣r∣<1|r| < 1. Then rn→0r^n \to 0 as n→∞n \to \infty, so the partial sums settle on a limit. If ∣r∣≥1|r| \ge 1 the series diverges\textbf{diverges}.

∣r∣<1⇒convergent|r| < 1 \Rightarrow \text{convergent}
Otherwise the series diverges (no finite sum).
2

The formula

For a convergent geometric series, the sum to infinity is the first term over 11 minus the ratio.

S∞=a1−r,∣r∣<1S_\infty = \dfrac{a}{1 - r}, \quad |r| < 1
Comes from SnS_n as rn→0r^n \to 0.
Example 1Find the sum to infinity of 8, 4, 2, 1, …
1a=8a = 8, r=12r = \tfrac{1}{2}.
2S∞=81−12=80.5S_\infty = \dfrac{8}{1 - \tfrac{1}{2}} = \dfrac{8}{0.5}.
Answer
1616
3

Using S∞

Questions may give S∞S_\infty and a term and ask for aa or rr. Substitute into the formula and solve. Always check the convergence condition ∣r∣<1|r| < 1 holds.

Tip — No finite sum to infinity exists unless |r| < 1 — state this condition in your answer.

Formula recap

S∞=a1−rS_\infty = \dfrac{a}{1 - r}
Sum to infinity (|r| < 1).
∣r∣<1⇒convergent|r| < 1 \Rightarrow \text{convergent}
Convergence condition.
∣r∣≥1⇒divergent|r| \ge 1 \Rightarrow \text{divergent}
No finite sum.

Common mistakes to avoid

Using S∞ when |r| ≥ 1.
The sum to infinity only exists for |r| < 1.
Writing S∞ = a/(r − 1).
It is a/(1 − r).

Key takeaways

  • A geometric series converges iff |r| < 1.
  • S∞ = a/(1 − r) for a convergent series.
  • State the |r| < 1 condition when using it.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. A geometric series converges when:

    • ∣r∣<1|r| < 1
    • ∣r∣>1|r| > 1
    • r=1r = 1
    • a<1a < 1
    Show answer and explanation

    Answer: ∣r∣<1|r| < 1

    The ratio must shrink terms.

  2. The sum to infinity is:

    • a1−r\dfrac{a}{1 - r}
    • ar−1\dfrac{a}{r - 1}
    • a(1−rn)1−r\dfrac{a(1-r^n)}{1-r}
    • arn−1ar^{n-1}
    Show answer and explanation

    Answer: a1−r\dfrac{a}{1 - r}

    First term over 1 − r.

  3. The sum to infinity of 8, 4, 2, 1, … is:

    • 1616
    • 1515
    • 88
    • ∞\infty
    Show answer and explanation

    Answer: 1616

    8 / (1 − ½) = 16.

  4. If ∣r∣≥1|r| \ge 1, the series:

    • Diverges
    • Converges
    • Sums to a
    • Sums to 0
    Show answer and explanation

    Answer: Diverges

    No finite sum.

  5. As n→∞n \to \infty with ∣r∣<1|r| < 1, rn→r^{n} \to

    • 00
    • 11
    • ∞\infty
    • r
    Show answer and explanation

    Answer: 00

    That gives the S∞ formula.

  6. A series with a=10a = 10, r=0.2r = 0.2 has S∞ =

    • 12.512.5
    • 5050
    • 88
    • 22
    Show answer and explanation

    Answer: 12.512.5

    10 / 0.8 = 12.5.

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