A substituted series and its endpoints

Choose your method, connect the parts and explain your conclusions. These original questions include written reasoning and sketches as well as exact answer checks.

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A-level prerequisite. Course filters use selected objective mappings; none of these tasks establishes complete paper coverage.

Before you start

Maclaurin series. Read the relevant method, then decide which parts you can solve without hints.

Check a misconception first · Try a connected problem

Read an original example question

A negative-power binomial series is used to expand $[1-2(x-2)]^{-2}$ .

Part a

Find the first three terms in powers of x − the centre.

Part b

State the lower and upper boundaries of the convergence interval.

Part c

Determine whether either boundary is included. Justify your answer.

Model solution and review criteria

Part a

With u = 2(x − 2), (1−u)⁻² = 1 + 2u + 3u² + … . The terms are 1 + 4(x−2) + 12(x−2)².

Identify the substituted variable.
Apply the negative-power coefficients.
Keep the centre in the expansion.

Part b

Require |2(x−2)| < 1. The boundaries are $\frac{3}{2}$ and $\frac{5}{2}$ .

Solve the inequality for x.
Do not simply state |x| < 1.

Part c

At u = ±1 the terms have magnitude n+1 and do not tend to zero. Neither boundary is included.

Check the series terms, not only the original function.
Explain divergence at both boundaries.

Compare your own reasoning. No automatic examination marks are awarded.