Approximation / Leibniz

Can you help me solve this problem using Leibniz theorem?

How many terms are needed in the taylor expansion of cos x to computer cos x for |x|<0.5 accurate to 12 decimal places (rounded)? The answer in the book is "at least 7 terms". But I need hints. I started with x^(2[n+1])/(2[n+1])! < 10^-12, but I didn't know how to continue.

Originally Posted by chadchoud

Can you help me solve this problem using Leibniz theorem?

How many terms are needed in the taylor expansion of cos x to computer cos x for |x|<0.5 accurate to 12 decimal places (rounded)? The answer in the book is "at least 7 terms". But I need hints. I started with x^(2[n+1])/(2[n+1])! < 10^-12, but I didn't know how to continue.

I think the folks at artofproblemsolving.com would be better help. I'll try, but I'm no match.

I would say you are on the right path except I would use (n-1) for the nth term, rather than (n+1). Your expression looks like absolute value of the term number (n+2).

Next, I would susbstitute 0.5 for x. Realize that for the expansion f(x)=f(-x) , and that the absolute value of each term increases as x increases towards the maximum for x=0.5.

thank you