How to Find an Upper Bound for Derivative Error in Lagrange Formula

[brunie]

Consider the function f(t) = ln (1 +2x)

Give a formula for f^(n) (x) [**the nth derivative] valid for all n >= 1 and find an upper bound for | f^(n) (x) | on the interval -0.25 <= x <= 0.25.
[ the error ].

I found the nth derivative to be

f^(n) (x) = (-1)^n+1 * 2^n /n * n!
--------------------------------
(1 + 2x)^n

so for
first derivative = 2 / 1+2x
second " " = -4 / (1+2x)^2
third " " = 16 / (1+2x)^3
etc.

now for the error i kno there is a lagrange error bound equation for taylor polynomials, but the question isn't for the taylor polynomial, only the "derivative generator"

i kno the max |f^(n+1)| <= M on an interval

so i just need help dealing with only the derivative error and i also want to kno how to find the M value in general with Taylor polynomials (not part of the above question)

where |f(x) - P_n(x)| <= M / (n+1)! * |x-a|^(n+1)
for interval between a and x

 

[gammamcc]

You are essentially done when you minimize the denominator in absolute value. Note, only "an upper bound" is asked for: You have room to over estimate.