Using the Maclaurin series to approximate $f(0.1)$ for $f(x)=(3+e^{2x})^{0.5}$

Question

I was tasked to use the Maclaurin series to calculate $f(0.1)$ of $f(x)=(3+e^{2x})^{0.5}$.

I got the Maclaurin expansion of $p_2(x) = \sqrt{3} + 4x +5x^2$ into which I plugged $0.1$ to yield $0.44+\sqrt{3}$. I also estimated the error but that gave me $$E< \frac{ 12 e^{0.2} +16 e^{0.4}} { 6000}$$ which is far less then what is measured.

What did I do wrong here?

Answer 1

Checking your expansion I got:

$$ 2+\frac{x}{2}+\frac{7 x^2}{16}+...+$$

for the Maclaurin expansion. This agrees well (when plugging in x = .1 )with the actual answer and is less than your error estimate which is just as it should be.