from here I start trying to find the notation to express this. I can get as far as:
As you can probably tell from the '...' that is the part I am having difficulty with. Hopefully my conundrum makes sense.
The problem is how to represent the sequence
The easiest way to do this is using the notation, called "product notation" or "pi notation." This is very similar to the notation (called sum or sigma notation), but instead of a sum, it is a product, i.e. you multiply all the terms together.
Using this notation, we can write and for .
So the fancy way to write your Taylor polynomial is
But to be honest, this is a little bit too much work for you to get all the indexes lined up (hopefully I didn't make a mistake) and for your readers to decode the dense symbols. Readability is often more important than getting things really compact. So it would probably be better for everyone to just write it as you had it originally:
As you can see, the second way takes up about the same amount of space as the first, but the second is actually a lot easier to understand, don't you think? Which would you rather see on a Calculus exam?