<s>[INST] Checking a newtonian binomial expansion I am trying to check that the following expansion of $\sqrt{1-x}$ using Newton's Binomial Theorem "appears correct by squaring both sides"
$$(1-x)^{1/2}=1-\frac{x}{2}-\frac{x^2}{8}-\frac{x^3}{16}-\frac{5x^4}{128}+...$$
then to do the check, I got the following:
$$(\sqrt{1-x})^2=(1-\frac{x}{2}-\frac{x^2}{8}-\frac{x^3}{16}-\frac{5x^4}{128}+...)^2$$
$$\Rightarrow 1=1+\frac{25x^8+80x^7+224x^6+896x^5-16384x}{16384}$$
 [/INST] Your last line is wrong. Expanding the first few terms on the right-hand side:
$$\left(1-\frac{x}{2}-\frac{x^2}{8}-\frac{x^3}{16}-\ldots\right)^2=1-2\cdot\frac{x}{2}+\frac{x^2}{4}-2\cdot\frac{x^2}{8}+\frac{x^4}{64}+2\cdot\frac{x^3}{16}-2\cdot\frac{x^3}{16}+\ldots$$
Everything cancels apart from $1-2\cdot\frac{x}{2}=1-x$.
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