Math Is Fun Forum

Sorry, when I first saw your question, I thought you were taking the limit as x goes to infinity for some reason.  Notice that taking x to infinity in your definition of e is the same is taking x to zero in mine, in which both cases we get e° = 1, as expected.

However, using

seems a bit more intuitive that e° = 1.

Another interesting fact is that

The letter you pick for your index variable doesn't matter.  Sometimes j or k is used instead of i.  But your line ∑ (2L+1) = ∑ (2n+1) = n² is incorrect.

The variable n represents the the largest number you are summing to (from 0 to n, perhaps).  The i is just a place holder to let you know how far along your summation you are.

The limit you referred to is 1.  I think you were actually inquiring about this:

That's just one way of defining e.  The trick isn't to prove that the limit as x goes to infinity is e; rather, it is to show that this limit exists and agrees with the other definitions that e possesses.

Check out http://web01.shu.edu/projects/reals/numseq/s_euler.html.

For differential equations, make sure you have a good background in Trigonometry, Algebra, and Geometry.  Of course you will need Calculus to take derivatives and integrate and such, but a solid foundation in TAG line of thinking (see what I did there?  neat, huh?) is important for your success.  Good luck!

Ok!  I posted this question on Yahoo! answers, and I got a reply there as well.  I took Rick's advice and combined it with the reply from Yahoo to get the following.  Please let me know if there are any mistakes.  Oh, and thanks for the warning about the chain rule!!!

Thanks for the tip about /displaystyle