D-Operator Homework Help: Understanding D^2 - 3D/5 Solved Example

[DryRun]

I'm reviewing my notes and i can't understand this part:

http://s1.ipicture.ru/uploads/20111202/OlsTV2UV.jpg

I have enclosed each corresponding term in the same coloured box. I've reviewed this solved example several times now but can't figure out how the result was obtained in the last line. I expanded each term (D^2 - 3D)/5 according to its power but ended up with terms exceeding powers of 3.

 

[I like Serena]

Hi sharks!

This is an application of Taylor's theorem.
In this particular case it means that:
1/(1+x)=1-(1/x)+(1/x)^2-...

It holds if |x| is less than 1, and also, in this particular case, it is the result of the corresponding geometric series.

 

[Fredrik]

This is an application of Taylor's theorem.
In this particular case it means that:
1/(1+x)=1-(1/x)+(1/x)^2-...

It holds if |x| is less than 1, and also, in this particular case, it is the result of the corresponding geometric series.

I think you meant
$$\frac{1}{1+x}=1-x+x^2-x^3+\cdots$$ Anyway, that explains the second line, but sharks appears to be asking about the third.

I've reviewed this solved example several times now but can't figure out how the result was obtained in the last line. I expanded each term (D^2 - 3D)/5 according to its power but ended up with terms exceeding powers of 3.

Yes, that's what the dots are for... There are infinitely many terms in the second line too. The last term before the dots in the second line is included to make sure that all terms with D³ will be included in the last line. Higher order terms in the second line (the terms represented by dots) don't contain any Dⁿ terms with n<4.

 

[I like Serena]

Blamey! Yes, that's what I meant.

 

[DryRun]

It's actually a Binomial series. And after some more digging around the net, i found the perfect example:
http://www.ucl.ac.uk/Mathematics/geomath/level2/series/ser84.html

Thanks for the help. I didn't know what it was, as i had just expanded the whole thing at first.

Cheers!

 

[I like Serena]

Oh, it's a specific occurrence of the binomial series for n=-1 instead of a general n.
Note that the binomial series is a special case of Taylor's theorem.

Cheers!