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10 cards from this deck
∣ax∣<1|ax| < 1∣ax∣<1 ensures the expansion converges and is valid.
∣x∣<1/8|x| < 1/8∣x∣<1/8 for the expansion of 1+8x\sqrt{1 + 8x}1+8x to hold.
∣x∣<17|x| < \frac{1}{7}∣x∣<71 makes the expansion of 41−7x3\frac{4}{\sqrt[3]{1 - 7x}}31−7x4 valid.
Rewrite as (1+(2/3)x)−2(1 + (2/3)x)^{-2}(1+(2/3)x)−2 and then apply the binomial.
The first term is 111 for the expansion of (1−2x)−1/2(1 - 2x)^{-1/2}(1−2x)−1/2.
The coefficient is 3/23/23/2 after binomial series expansion.
∣x∣<1|x| < 1∣x∣<1 ensures convergence of the series.
∣ax∣|ax|∣ax∣ must be less than 1 for the expansion to be valid.
∣x∣<3/2|x| < 3/2∣x∣<3/2 keeps the expansion valid; powers of xxx converge.
Factor out 3−23^{-2}3−2, then apply binomial expansion.
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