Taylor Series Remainder Theorem

[stupidmonkey]

1. Prove that the MacLaurin series for cosx converges to cosx for all x.

Homework Equations

Ʃ(n=0 to infinity) ((-1)^n)(x^2n)/((2n)!) is the MacLaurin series for cosx
|Rn(x)|\leqM*(|x|^(n+1))/((n+1)!) if |f^(n+1)(x)|\leqM
lim(n->infinity)Rn=0 then a function is equal to its Taylor series within the interval of convergence.

3. The Attempt at a Solution . I already know what the solution is but I can't figure out why we are using this solution.
First you get: |f^(n+1)(x)|\leq1 because |cosx| and |sinx| functions are always bounded by 1
Then you write |Rn(x)|\leq(|x|^(n+1))/((n+1)!)
lim as n->infinity of (|x|^(n+1))/((n+1)!)=0, therefore lim as n-> infinity of |Rn|=0 so the MacLaurin series of cosx converges to cosx for all x.
My question
Why don't we just write Rn = \Sigma from (i=n+1 to infinity) ((-1)^i)*(x^2i)/((2i)!) and we can see from this that the limit as n->infinity makes Rn go to zero, therefore cosx converges to cosx? Why did we have to go through all of that other stuff?

 

[Office_Shredder]

If you write that the remainder is equal to the terms that you didn't sum up yet, then you're assuming that the infinite series is equal to cos(x) in the first place.

There are functions where the Taylor series converges, but does not converge to the function, so you can't assume that the function is equal to the Taylor series and just calculate the interval of convergence