# Thread: Properties of Power Series

1. ## Properties of Power Series

Hi!

I'm taking DFQ and we are learning about solving DE's using the series method.
I was wondering what the some properties for multiplying 2 power series' are?

Specifically I need to know how to square an infinite power series. I know I cannot just take the square into the inside of the series, but I cannot find anything online about a formula or method.

Thanks!

2. ## Re: Properties of Power Series

Originally Posted by beatrixb
Hi!

I'm taking DFQ and we are learning about solving DE's using the series method.
I was wondering what the some properties for multiplying 2 power series' are?

Specifically I need to know how to square an infinite power series. I know I cannot just take the square into the inside of the series, but I cannot find anything online about a formula or method.

Thanks!
Follow the same rules of expanding that you already know, i.e. first term x everything in the second bracket + second term x everything in the second bracket + third term x everything in the second bracket + ...

3. ## Re: Properties of Power Series

Originally Posted by Prove It
Follow the same rules of expanding that you already know, i.e. first term x everything in the second bracket + second term x everything in the second bracket + third term x everything in the second bracket + ...
Ahh okay so if it were two different series' (A & B) I would get A^2+2AB+B^2?
And with only one series (A) it is A^2+2A^2+A^2 which would then equal 4A^2?

4. ## Re: Properties of Power Series

Originally Posted by beatrixb
Ahh okay so if it were two different series' (A & B) I would get A^2+2AB+B^2?
And with only one series (A) it is A^2+2A^2+A^2 which would then equal 4A^2?
No, I mean

$\displaystyle \displaystyle (a_1 + a_2 + a_3 + \dots)(b_1 + b_2 + b_3 + \dots)\\ = a_1(b_1 + b_2 + b_3 + \dots) + a_2(b_1 + b_2 + b_3 + \dots) + a_3(b_1 + b_2 + b_3 + \dots) + \dots$

5. ## Re: Properties of Power Series

Originally Posted by beatrixb
Hi!

I'm taking DFQ and we are learning about solving DE's using the series method.
I was wondering what the some properties for multiplying 2 power series' are?

Specifically I need to know how to square an infinite power series. I know I cannot just take the square into the inside of the series, but I cannot find anything online about a formula or method.

Thanks!
The formula for the product of two 'infinite sums' is due to Cauchy...

$\displaystyle \sum_{n=0}^{\infty} a_{n}\ \sum_{k=0}^{\infty} b_{k} = \sum_{n=0}^{\infty} \sum_{k=0}^{n} a_{k}\ b_{n-k}$ (1)

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$