I know that the Maclaurin series expansion for $$\frac{1}{\sqrt{1+x^2}}$$ using binonmial coefficients is:
= 1 - $\frac{x^2}{2}$ + $\frac{3x^4}{8}$ - $\frac{5x^6}{16}$ + $\frac{35x^8}{128}$ + O(x^9)
However, when I try to use derivative rules, evaluating at a = 0, I get
f(0) = 1
f '(x) = $\frac{2x}{-2(1+x^2)^(3/2)}$ which = 0 at when a = 0
All other derivatives have x in the numerator, which also give coefficients of 0.
As a result, the Maclaurin series using derivatives would be:
1 + 0 + 0 + 0....
Why is this the case? I would expect that the Maclaurin series using binomial coefficients matches the Maclaurin series obtained by taking derivatives.
Check your derivatives again.
They are not all zero at a=0.
For example when you find your f''(x) using quotient rule, the derivative of the top will generate a non-zero value at a=0.