# representation of functions as power series

• Apr 18th 2009, 11:02 PM
twilightstr
representation of functions as power series
a) use differentiation to find a power series representation for f(x)= 1/(1+x)^2
what is the radius of convergence?
my answer: sigma from n =1 to infinity of (-1)^n[nx^(n-1)] R = 1 is this right. should n start at one or zero.
b) use part (a) to find a power series for f(x)= 1/(1+x)^3
my answer:n =1 to infinity of (-1)^n[n(n-1)x^(n-2)] is this right too R =1
c) use part (b) to find a power series for f(x)= x^2/(1+x)^3
my answer: n =1 to infinity of (-1)^n[n(n-1)x^(n)] is this right too R =1
is this right too?
• Apr 19th 2009, 01:07 AM
Moo
Hello,
Quote:

Originally Posted by twilightstr
a) use differentiation to find a power series representation for f(x)= 1/(1+x)^2
what is the radius of convergence?
my answer: sigma from n =1 to infinity of (-1)^n[nx^(n-1)] R = 1 is this right. should n start at one or zero.

It's correct. It's better to make it start at n=1. Although it's not a big problem to start at n=0, because the n=0 term is 0.

Quote:

b) use part (a) to find a power series for f(x)= 1/(1+x)^3
my answer:n =1 to infinity of (-1)^n[n(n-1)x^(n-2)] is this right too R =1
Same as above. But following your logic, it should have been n=2 here

Quote:

c) use part (b) to find a power series for f(x)= x^2/(1+x)^3
my answer: n =1 to infinity of (-1)^n[n(n-1)x^(n)] is this right too R =1
is this right too?
yes (n=2)
• Apr 19th 2009, 01:44 PM
twilightstr
however in the answer key (b) and (c) are multiplied by 1/2. why is that?
• Apr 19th 2009, 11:25 PM
Moo
Quote:

Originally Posted by twilightstr
however in the answer key (b) and (c) are multiplied by 1/2. why is that?

Oh no, I'm sorry !! I forgot that one :(

When you differentiate $\displaystyle \frac{1}{(1+x)^2}=(1+x)^{-2}$, what do you get ?
Recall that the derivative of $\displaystyle [u(x)]^n$ is $\displaystyle u'(x) \cdot {\color{red}n} \cdot [u(x)]^{n-1}$
There's also a minus sign that should go with the 1/2 ;)