# Rate of Convergence

• Sep 30th 2012, 11:58 AM
sjmiller
Rate of Convergence
Hi All,

I've been trying to figure out how to do the question in the attached file for quite a few days now, and I think I've got most of it figured out. I need to find the rate of convergence of the function as h approaches 0. I have attached both the question and my work. I'm down to the final inequality and have no idea how to find O(h^2) which is the answer to this question.

The function is lim h->0 (sinh+hcosh)/h

Any help would be appreciated.

Thank you.
• Sep 30th 2012, 01:47 PM
MaxJasper
Re: Rate of Convergence
Hint:

Series expansion of each function near 0 is:

$\displaystyle \text{Sin}(x)/x = 1-\frac{x^2}{6}+\frac{x^4}{120}-\frac{x^6}{5040}+O(x)^8$ [updated]

$\displaystyle \text{Cos}(x) = 1-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}+O(x)^8$
• Sep 30th 2012, 01:57 PM
sjmiller
Re: Rate of Convergence
Quote:

Originally Posted by MaxJasper
Hint:

Series expansion of each function near 0 is:

$\displaystyle \text{Sinh}(x)/x = 1+\frac{x^2}{6}+\frac{x^4}{120}+\frac{x^6}{5040}+O (x)^8$

$\displaystyle \text{Cos}(x) = 1-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}+O(x)^8$

Hi, it is sin(x)/x not sinh(x)/x so I'm a bit confused :S
• Sep 30th 2012, 02:46 PM
MaxJasper
Re: Rate of Convergence
See update