Power Series

Printable View

• December 4th 2006, 06:30 AM
TreeMoney
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!!!!
• December 4th 2006, 08:09 AM
CaptainBlack
Quote:

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:

$
f(x)=\sum_0^{\infty} a_n x^n
$

is sufficiently well behaved then:

$
f'(x)=\sum_1^{\infty} a_n n x^{n-1}
$

and in general:

$
f^{(k)}(x)=\sum_k^{\infty} a_n \frac{n!}{(n-k)!} x^{n-k}
$

so:

$
f^{(k)}(0)=a_k \, {k!}
$

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 $x=0$)

RonL
• December 4th 2006, 08:18 AM
galactus
Quote:

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 $tan^{-1}(x^{2})-(tan^{-1}(x))^{2}$ and you get an end term of $\frac{2}{3}x^{4}$

When you divide by $x^{4}$, you get $\frac{2}{3}$

Which is the limit as x approaches 0.
• December 6th 2006, 09:41 AM
TreeMoney
Still don't get it
I tried doing part A and am still clueless. Can someone help me out. TIA