Sum to infinity, and when it exists

In 7 of 20 recent exams
Paper 1 or 2
1 to 5 marks

How it is set in the exam

Paper 2 (calculator) mostly (5 of its 7 exam versions), 1–5 marks a part: a context with infinitely many terms (savings that never reach a target, a bouncing ball, a damped motion) or an infinite sum written with $\sum$ . On Paper 1 (no calculator) (2 versions) it is exact, sometimes a long Section B question: find $r$ in terms of $x$ (or from log terms), then when $S_\infty$ exists, then solve for $x$ . Common (7 of 20 exam versions, 8 parts).

Spot it in the exam

Exam phrases to spot (3)

“sum to infinity”
“the series converges”
“find the values of $x$ for which the sum to infinity exists”

In disguise (8)

"the total distance travelled before the ball comes to rest"; "the total before it stops": infinitely many terms, so $S_\infty$
a series with $x$ in the terms, where $r$ is an expression: linear ( $1 + 2x + 4x^2 + \cdots$ ) or not ( $x + 3 + \frac{9}{x} + \cdots$ has $r = \frac{3}{x}$ )
an infinite sigma: $\sum_{r=1}^{\infty} 6(0.5)^r$ (first term at $r = 1$ : $3$ )
"the sum to infinity is three times the first term" (an equation in $r$ only)
"show that the total never reaches…": no $n$ works, and the proof is $S_\infty$ . Savings of 400 in the first month, then 5% less each month, have $S_\infty = \frac{400}{1 - 0.95} = 8000$ . The terms are positive, so every $S_n$ is below $8000$ : the total never reaches $8500$ . Calculate $S_\infty$ first; a few partial sums prove nothing.
log terms such as $\ln x^9$ , $\ln x^p$ , $\ln x$ : bring the powers down with the log laws, factor out $\ln x$ , and the coefficients form the sequence (logs of a sequence)
a damped motion that keeps passing back through its starting point, each swing's greatest distance the same fraction of the one before: every distance is covered out and back, so the total is $2S_\infty$ , with the first term doubled too (a dropped ball's first fall is covered only once)
"find the values of $k$ for which…": same wording as the discriminant questions, different condition. See.

Worked example

When it exists, then solve for $x$

Paper 1 (no calculator)

Question

The first two terms of a geometric series are $4$ and $12x - 8$ .
(a) Find the values of $x$ for which the sum to infinity exists. [4]
(b) Given that the sum to infinity is $5$ , find $x$ . [3]

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Steps

Find $r$ by dividing the second term by the first. It may be an expression in $x$ .
Exists?
Solve $\lvert r \rvert < 1$ , i.e. $-1 < r < 1$ , for $x$ . If $r$ is linear in $x$ this is one interval. If $x$ is in a denominator ( $r = \frac{3}{x}$ ) it is $\lvert x \rvert > 3$ : two rays. With a trig $r$ , sketch it: there can be several intervals.
Value?
$S_\infty = \frac{u_1}{1 - r}$ .
$S_\infty$ given?
Form the equation and solve. With a second unknown this is the Find $u_1$ and $d$ (or $r$ ) from two facts path, with $S_\infty$ as one fact. Check every solution satisfies $\lvert r \rvert < 1$ and reject any that don't, saying why.

Common mistakes

Using $S_\infty$ when $\lvert r \rvert \geq 1$ .
The formula still gives a number, but it means nothing.
Solving $\lvert r \rvert < 1$ as $r < 1$ only.
$-1 < r$ matters too: it is half the condition.
Solving $\left\lvert \frac{3}{x} \right\rvert < 1$ as $x > 3$ only.
It is $\lvert x \rvert > 3$ : $x < -3$ or $x > 3$ . Test a negative value such as $x = -6$ to see it.

More mistakes (2)

Not checking roots
against $\lvert r \rvert < 1$ when $S_\infty$ is given, or rejecting without saying why.
Bouncing ball
forgetting to double the bounce heights, or doubling the first drop too.

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