Question

g

Answer 1

You must match the indeces in order to add the arguments. It is sometimes necessary too peel off a few terms.

Answer 2

Maybe. It's a little like the old algebra problem...

Three consecutive integers sum to 75...

Should you define x, x+1, and x+ 2?
Sometimes, it is more convenient to define x-1, x, x+1.

For the indeces, you have 0, 1, and 2. It doesn't much matter where you match them up, except that in some situation one or the other may be more convenient. It's a choice, not a requirement.

Answer 3

Yup. Good work. Power series solutions can be a bit tedious. I'm guessing you already discerned that. :-)

Answer 4

Factored?

What have you against pulling out two terms?