Thread: Power Series

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!!!!

Originally Posted by TreeMoney

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.)

If:



is sufficiently well behaved then:



and in general:



so:



Which should be sufficient to do part a).

(In this case sufficiently well behaved is satisfied by the series being absolutely
convergent in some neighbourhood of )

RonL

b) What is limx−>0 ((arctan(x^2)−(arctan (x))^2)/x4) ? (Do not use l’Hospital! - it would be a waste of time.)

It turns into a monstrisity with L'Hopital, though, it can be done.

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.

I tried doing part A and am still clueless. Can someone help me out. TIA