Discover the Maclaurin Series for f(x) with Derivative Calculations

[defang]

My question is as follows: Let f (x) = (1+x)^(1/2) – (1-x)^(1/2). Find the Maclaurin series for f(x) and use it to find f ^5 (0) and f ^20 (0).

I got: X + Riemann Sum { [ (-1)^(n-1) 1x3x5**x(2n-3) ] / (2^n) x n！} X^n (after combining two Riemann Sums together). And I got (7！5！) / 16 5！ = 315. However, I tried to check my answer by taking derivative 5 times, and I got 105/16. Can anyone tell me what I did wrong? thanks

 

[defang]

?

Is this question unsolvable? Or is it .....? Please, somebody gives me some advice....

 

[Zurtex]

This really should be put in the homework section, but anyway, I can't tell you what you have done wrong but it'd probably be much easier to try and work out the nth derivative of the series, or at least the nth derivative at x = 0.

 

[steven187]

hello there

if I were you I will try to make the problem look more simpler, why don't you split the function, like f(x)=g(x)-h(x) where
h(x)=(1-x)^(1/2) and
g(x)=(1+x)^(1/2)
find the maclarin series for each h(x) and g(x) then find the maclarin series for f(x)
and as for the last parts f^n(x)=g^n(x)-h^n(x) this should most likely work

steven