# Thread: representation of functions as power series

1. ## 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?

2. Hello,
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.

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

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)

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

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