Sequences Question (1 Viewer)

[Shadowdude]

So I'm doing some power solutions to DEs and I get expressions like this:



with



given.

And the solution at the 'back of the book' is:



And if we compare, we see that the two solutions match up (well, at least appear to do so after four terms)


My question is: How do I go from my power series above - to the neat summation at the bottom?

Is there a trick? Or is the solution provided just a quick way to summarise the series and I'm not actually required to do such a thing, because my expansion is equivalent anyway...


any and all help is appreciated

thanks in advance!


edit:

not sure if this is relevant, but the recurrence used to achieve the expansion is



for k = 0, 1, 2, ...

 

[seanieg89]

It is an easy recurrence to solve...



And I would recommend doing this (solving the recurrence) whenever possible in such questions.

 

[Shadowdude]

ohhhh... okay. i knew i was missing something.

my bad

Thanks seanieg!

note to self: don't do maffs late at night.


edit: then i realise i haven't done any work at all on recurrences of that nature

*sigh*


Yeah, so I looked through my lecture notes for Discrete Maths where we covered recurrences. We didn't learn what to do for non-constant coefficient recurrences. oh well

 

[seanieg89]

This is why mechanically learning things is not as important as being able to spot patterns. Just look at the product expression for A_2k when you unravel using the recurrence relation and express it in terms of factorials. This sort of thing is done in some hsc questions.