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Binomial series expansion

Binomial series expansion

IB Year 6 | Grade 12

Obtain the first three non-zero terms of the expansion of

  1. \((1+2x)^{-1}\)
  2. \[\frac{x}{\sqrt{1+2x}}\]

in

  1. ascending powers of \(x\)
  2. descending powers of \(x\).

In each case, find the range of \(x\) for which the expansion is valid.

Solution:

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Answer:(a)(i) \(1-2x+4x^2-\cdots\), \(|x|<\frac12\). (a)(ii) \(\frac1{2x}-\frac1{4x^2}+\frac1{8x^3}-\cdots\), \(|x|>\frac12\). (b)(i) \(x-x^2+\frac32x^3-\cdots\), \(-\frac12<x\leq\frac12\). (b)(ii) \(\frac{x^{\frac12}}{\sqrt2}-\frac{x^{-\frac12}}{4\sqrt2}+\frac{3x^{-\frac32}}{32\sqrt2}-\cdots\), \(x\geq\frac12\). In (b) the endpoint \(x=\frac12\) follows from a separate convergence check; the usual strict binomial ranges omit that endpoint.

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