# Thread: I need power series help and I don't get when to change the index

1. ## I need power series help and I don't get when to change the index

The problem:

arctan(x/3)

I assume you derive it: 1/(1+(x/3)²)

then put it in the proper formation: 1/(1- (-x/3)²)

so then I get the sigma of -1^2n -x/3²^n

and then I have to take the integral?

I just don't get what to do, like the proper formation. Why do I have to keep the -1 and -x/3 seperate if they are both to the 2n power?

And in an unrelated note, when do you have to change the index? I don't get that AT ALL.

FML. thanks

2. ## Re: I need power series help and I don't get when to change the index

Originally Posted by skinsdomination09
The problem:

arctan(x/3)

I assume you derive it: 1/(1+(x/3)²)

then put it in the proper formation: 1/(1- (-x/3)²)

so then I get the sigma of -1^2n -x/3²^n

and then I have to take the integra
I really have no idea what that says.

But look at this webpage.
Does that help?

3. ## Re: I need power series help and I don't get when to change the index

Originally Posted by skinsdomination09
The problem:

arctan(x/3)

I assume you derive it: 1/(1+(x/3)²)

then put it in the proper formation: 1/(1- (-x/3)²)

so then I get the sigma of -1^2n -x/3²^n

and then I have to take the integral?

I just don't get what to do, like the proper formation. Why do I have to keep the -1 and -x/3 seperate if they are both to the 2n power?

And in an unrelated note, when do you have to change the index? I don't get that AT ALL.

FML. thanks
First of all, \displaystyle \displaystyle \begin{align*} \frac{1}{1 - \left( -\frac{x}{3} \right) ^2} \end{align*} is NOT the same as $\displaystyle \displaystyle \frac{1}{1 + \left( \frac{x}{3} \right) ^2}$. You should be writing it as \displaystyle \displaystyle \begin{align*} \frac{1}{1 - \left[ - \left( \frac{x}{3} \right) ^2 \right]} \end{align*}.

Now the easiest thing to do is to remember that an infinite geometric series $\displaystyle \displaystyle \sum_{n = 0}^{\infty} a\,r^n = \frac{a}{1-r}$, which is convergent when |r| < 1. Can you see that what you have looks like this?