Legendre differential equation- power series

[Jenkz]

Homework Statement

http://mathworld.wolfram.com/LegendreDifferentialEquation.html

I have a question about how the website above moves from one equation to another etc.

1./ Equations (4), (5) and (6)

When differentiating (4) to (5) shouldn't the the limit be from n=1, which means (5) should be the sum of (n+1) a_(n+1) x^n from n=0 to infinitely?

Same with (6), shouldn't the series expansion on the website be from n=2 not n=0?

2./ Equations (8) and (10)

Why does the first series term suddenly change from being a limit from n=0 to n=2 ? And so why does the second term in equation (10) not also change from having a limit from n=0 (in equation (8) ) to n=2 ?


I'm sorry if my questions are a bit confusing or badly explained. Help will be much appreciated :) Thanks.

 

[tiny-tim]

Hi Jenkz!

When differentiating (4) to (5) shouldn't the the limit be from n=1, which means (5) should be the sum of (n+1) a_(n+1) x^n from n=0 to infinitely?

Same with (6), shouldn't the series expansion on the website be from n=2 not n=0?

Because it's best not to change the limits unless you have to …

that makes a mistake less likely, and easier to spot!

2./ Equations (8) and (10)

Why does the first series term suddenly change from being a limit from n=0 to n=2 ? And so why does the second term in equation (10) not also change from having a limit from n=0 (in equation (8) ) to n=2 ?

Because this is one case where we do have to change the limits …

to get the first term in (12) from the first term in (8), we need to change both the parameter n and the limits … it's best to change only one at a time (to avoid mistakes!), so in the first term in (10), we change the limits, and then in the first term in (12) we change the parameter.

 

[Jenkz]

Thank you!
But I'm still wondering why we need to change the limits? I understand how the limits are change. But why does it need to be changed to n=2?

And why doesn't the second term in (10) not also need to have it's limit changed?

 

[tiny-tim]

But why does it need to be changed to n=2?

And why doesn't the second term in (10) not also need to have it's limit changed?

Because we need all the terms in (12) to be xⁿ,

(so we can divide by xⁿ and get (14))

and in (10) and (11) the first term was only x^n-2, so we had to bump it up a little!

(but the second term was already xⁿ, so we left it alone )

 

[Jenkz]

Ohhh I see, thanks again :)