## Exam-Style Question on Maclaurin Series## A mathematics exam-style question with a worked solution that can be revealed gradually |

Question id: 697. This question is similar to one that appeared on an IB AA Higher paper in 2021. The use of a calculator is allowed.

(a) Write down the first three terms of the binomial expansion of \( (1 - a)^{-1} \) in ascending powers of \( a \).

(b) By using the Maclaurin series for \( \dfrac{\sin x}{x} \) and the result from part (a), find the Maclaurin series for \( x\text{cosec} x \) up to and including the term in \( x^4 \).

(c) By using the Maclaurin series for \( \arctan x \) and the result from part (b) find:

$$ \lim_{{x \to 0}} \left( \frac{x \arctan 2x}{x\text{cosec}x - 1} \right) $$

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