How do I get insight into equations with the $\sum$ notation without actually expanding it for a specific n every time?

Question

I am reading Goldstein's Classical Mechanics and I've noticed there is copious use of the $\sum$ notation. He even writes the chain rule as a sum! I am having a real hard time following his arguments where this notation is used, often with differentiation and multiple indices thrown in for good measure. How do I get some working insight into how sums behave without actually saying "Now imagine n=2. What does the sum become in this case?" Is there an easier way to do this? Is there an "algebra" or "calculus" of sums, like a set of rules for manipulating them? I've seen some documents on the web but none of them seem to come close to Goldstein's usage in terms of sophistication. Where can I get my hands on practice material for this notation?

Answer 1

I remember that I completely lost my uneasines with sums after reading first several chapters of this book. Apart from being very educative, having lots of various excercises, and $\sum$ letter on its cover -- it is also very fun to read.

Answer 2

Just look at the formulas in the various derivations, abstract from them the operations that apparently have been going on to reaching the rhs from the lhs, keep a list of these rules, and take the recurring ones as permissions to do this sort of rearrangement or substitution.

If a text isn't too short of intermediate steps and doesn't have too many misprints, this reveals the hidden secrets in most calculations, not only sums.

Answer 3

Donald Knuth's book "The Art of Computer Programming" Volume 1 contains probably all the tricks of manipulating sums, you will ever need. If you just want to learn the material in Goldstein and find it hard, you should consider an alternative book with the same material.