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 Video by Zeeshan Zamurred Sequences and Series

Edexcel A level Maths: 3.5 Sum to InfinityWatch the full walkthrough 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∞

Convergence condition

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

∣ r ∣ < 1 \Rightarrow convergent

Otherwise the series diverges (no finite sum).

The formula

For a convergent geometric series, the sum to infinity is the first term over $1$ minus the ratio.

S_{\infty} = \frac{a}{1 - r}, ∣ r ∣ < 1

Comes from

S_{n}

r^{n} \to 0

a = 8

r = \frac{1}{2}

S_{\infty} = \frac{8}{1 - \frac{1}{2}} = \frac{8}{0.5}

Answer

16

Using S∞

Questions may give $S_{\infty}$ and a term and ask for $a$ or $r$ . Substitute into the formula and solve. Always check the convergence condition $∣ r ∣ < 1$ holds.

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

Formula recap

S_{\infty} = \frac{a}{1 - r}

Sum to infinity (|r| < 1).

∣ r ∣ < 1 \Rightarrow convergent

Convergence condition.

∣ r ∣ \geq 1 \Rightarrow 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

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