Recall that the power series of arctan x is the Sum from 0 to infinity = 0(−1)^n (x^(2n+1) / (2n+1)) , with a radius of convergence of 1.
a) What is the value of the sixteenth derivative of arctan x at x = 0? What is the value of the seventeenth derivative of arctan x at x = 0? (Hint: look at the term of that degree in the power series.)
b) What is limx−>0 ((arctan(x^2)−(arctan (x))^2)/x4) ? (Do not use l’Hospital! - it would be a waste of time.)
This is a homework problem and i have no clue how to even go about starting it. Can someone help me out. TIA!!!!
It turns into a monstrisity with L'Hopital, though, it can be done.b) What is limx−>0 ((arctan(x^2)−(arctan (x))^2)/x4) ? (Do not use l’Hospital! - it would be a waste of time.)
What they're getting at is Taylor series.
Expand the Taylor series for and you get an end term of
When you divide by , you get
Which is the limit as x approaches 0.