Determine the rate of convergence of each sequence and numerically determine which of the following sequences approaches 1 faster.
$$\lim_{x\to0} \frac{ sinx^2}{x^2} versus \lim_{x\to0} \frac{(sinx)^2}{x^2}$$ **
I just found out the rate of convergence for this sequence
$\lim_{x\to0} \frac{ sinx^2}{x^2}$.
For this,
I used Taylor's theorem
$\frac{sinx^2}{x^2} = 1- \frac{x^4}{6} \sin £ $
for some £ between 0 and x.
Then I found the rate of convergence to be $ O(x^4)$.
Just need help to find rate of convergence of other sequence.
Textbook i am using:- Brain Bradie
A friendly introduction to numerical analysis.
Best Answer
$\frac{sinx^2}{x^2} = 1- \frac{x^4}{6} \sin £ $
Rate of convergence of a sequence:- Let ${P_n}$ be a sequence that convergences to a number P. If there exists a sequence ${B_n}$ which converges to zero and a positive constant K such that $|P_n - P| \le |B_n|$ then the sequence convergences to P with rate of convergence $O(B_n)$.
So I found out $K= \frac{1}{6}$ and $B=x^4$. Similarly i need such steps for other sequence.