It looks to me like you forgot to take the square of the expansion... What you gave is an expansion for , isn't it?
You should consider using the formula .
Or painfully expand the square .
I wanted to find a Maclaurin series for . Here is what I did:
Spoiler:
But since , can't I directly use the formula without expanding at all? If so, could someone please show me how to do it. I have tried to compute but it doesn't appear to yield .
Damn it! I see what happened. Thank you, Laurent.
. This is where I stuck before as well when I was trying to calculate
Let me try it with :
We have Fabulous! I wouldn't really mind if someone showed me how to manipulate the above series into this one, though.
Since when do you have ? You just wrote , which is just the same mistake. There are many other terms in the square of a sum.
The product of two series is sometimes called "Cauchy product". If you expand the product, you get (look for "Cauchy product" on the web, if you've never seen this before). Substitute your value for , and now you understand why I mentioned pain (well, probably not that much actually). I guess the binomial formula would lead you back to the result obtained by the "fabulous" method.