Alternating Series Theorem (remainder formula)
The Alternating Series Estimation theorem states:
If s= from n=1 to infin. is the sum of a convergent, alternating series that satisfies:
a) 0 < b_n+1 < b_n
b) lim b_n -> 0 as n-> infin
|R_n|= s - s_n <= b_n+1
So the problem I'm given is this:
How many terms of the series do I need to add in order to find the sum to the indicated accuracy?
; error < 0.01
Okay, so I can use the theorem mentioned above because this series is both alternating and convergent (the positive part of the sum is decreasing for all n, and the lim 1/n! -> 0 as n-> infin, satisfying both conditions for convergence).
So... |R_n| <= b_n+1 < 1/100... right?
b_n+1 = 1/(n+1)! < 1/100
So how do I solve for n? Looking at the solution for this problem, it seems like the person kept guessing random values of n until (-2)^n/n! was less than 1/100, which I don't really understand. The answer ended up being 7. That is, the sum of 7 terms to keep within the given accuracy.
I guess I don't know how to do these kinds of problems when the "negative part of the series" is something other than (-1)^n.