1

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.

Kavita
  • 728
  • The other taylor series is $1-1/3x^2+2/45x^4+O(x^6)$ – Peter Nov 18 '16 at 13:29
  • What is meant by "numerically" ? The easiest way would be inserting the values $x=10^{-n}$ with $n=1,2,3,\cdots$ – Peter Nov 18 '16 at 13:32
  • I edited my question. Do you mean rate of convergence of other sequence is $O(x^6)$ – Kavita Nov 18 '16 at 13:42
  • No, I would say the rates are $O(x^2)$ and $O(x^4)$ – Peter Nov 18 '16 at 14:00
  • I need the steps in detail according to definition of rate of convergence of a sequence. Please – Kavita Nov 18 '16 at 14:02
  • I am not sure and do not want to claim something wrong. But it seems logical that the smallest non-constant power is the rate of convergence. No idea however, how this rate should be determined numerically. We could perhaps compare the rates numerically, but I do not know a reliable numerical method for this either. – Peter Nov 18 '16 at 14:04
  • No. This rate need not be shown numerically. I have to compare the rates numerically. Let me show you how I used the definition to find rate of one sequence. Give me some time to post the way. – Kavita Nov 18 '16 at 14:07
  • The problem is that one sequence could approach faster at the beginning, but on the long run, the other could approach faster. So, we never can be sure which function approaches faster overall. – Peter Nov 18 '16 at 14:13

1 Answers1

1

$\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.

Kavita
  • 728