Find a quadratic polynomial such that 2^x = P(x) + o(x²)

by RoRi

December 21, 2015

Find a quadratic polynomial $P(x)$ such that

$2^x = P(x) + o(x^2) \qquad \text{as} \qquad x \to 0.$

Since $2^x = e^{x \log 2}$ and

$e^x = 1 + x + \frac{x^2}{2} + o(x^2) \qquad \text{as} \qquad x \to 0,$

we have

$2^x = 1 + (x \log 2) + \left( \frac{\log^2 2}{2} \right) x^2 + o(x^2) \qquad \text{as} \qquad x \to 0.$

Therefore we may take

$P(x) = 1 + (\log 2)x + \frac{1}{2} x^2 \log^2 x .$

Stumbling Robot